ALBERT

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Alain Haraux〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We refine some previous sufficient conditions for exponential stability of the linear ODE 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉c〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉b〈/mi〉〈mo〉+〈/mo〉〈mi〉a〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mi〉b〈/mi〉〈mo〉,〈/mo〉〈mi〉c〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉a〈/mi〉〈/math〉 is a bounded nonnegative time dependent coefficient. This allows to improve some results on uniqueness and asymptotic stability of periodic or almost periodic solutions of the equation 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si4.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉c〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉g〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉c〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉g〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 satisfies some sign hypotheses. The typical case is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉g〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉b〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉a〈/mi〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mrow〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉a〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉.〈/mo〉〈/mrow〉〈/math〉 Similar properties are valid for evolution equations of the form 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si10.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉c〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉B〈/mi〉〈mo〉+〈/mo〉〈mi〉A〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉A〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mi〉B〈/mi〉〈/math〉 are self-adjoint operators on a real Hilbert space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mi〉H〈/mi〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mi〉B〈/mi〉〈/math〉 coercive and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉A〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 bounded in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si16.svg"〉〈mrow〉〈mi〉L〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉H〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 with a sufficiently small bound of its norm in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mo〉+〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi〉L〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉H〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Chun-Ku Kuo, Wen-Xiu Ma〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, the existence and non-existence of resonant multi-soliton solutions to two different (2+1)-dimensional Hirota–Satsuma–Ito (HSI) equations are explored. After applying the linear superposition principle we generate resonant multi-soliton solutions to the first HSI equation which appeared in the theory of shallow water wave. The conditions of real resonant multi-soliton solutions are revealed. The presented resonant multi-soliton solutions exhibit the inelastic collision phenomenon among the involved solitary waves. Particularly, upon choosing appropriate parameters, we demonstrate the characteristics of inelastic interactions among the multi-front kink waves both graphically and theoretically. Moreover, non-existence of resonant multi-soliton solution is considered for the generalized HSI equation via the linear superposition principle.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Jin Tao, Dachun Yang, Wen Yuan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉[〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mi〉s〈/mi〉〈/math〉 be a non-negative integer. In this article, the authors introduce the John–Nirenberg–Campanato space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉J〈/mi〉〈msub〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="script"〉X〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi mathvariant="script"〉X〈/mi〉〈/math〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 or any cube 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉⫋〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, which when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 coincides with the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉J〈/mi〉〈msub〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/math〉-space introduced by F. John and L. Nirenberg in the sense of equivalent norms. The authors then give the predual space of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉J〈/mi〉〈msub〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="script"〉X〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and a John–Nirenberg type inequality of John–Nirenberg–Campanato spaces. Moreover, the authors prove that the classical Campanato space serves as a limit space of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉J〈/mi〉〈msub〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="script"〉X〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo〉→〈/mo〉〈mi〉∞〈/mi〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Mohamed Karim Hamdani, Abdellaziz Harrabi, Foued Mtiri, Dušan D. Repovš〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work, we study the existence and multiplicity results for the following nonlocal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉-Kirchhoff problem: 〈span〉〈span〉(0.1)〈/span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉a〈/mi〉〈mo〉−〈/mo〉〈mi〉b〈/mi〉〈msub〉〈mrow〉〈mo linebreak="badbreak"〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈/msub〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfrac〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈/mrow〉〈/mfenced〉〈mi〉d〈/mi〉〈mi〉i〈/mi〉〈mi〉v〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉g〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mtext〉 in 〈/mtext〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mtext〉 on 〈/mtext〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉a〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 are constants, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉Ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 is a bounded smooth domain, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi〉C〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mover accent="false"〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈mo accent="true"〉¯〈/mo〉〈/mover〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mi〉λ〈/mi〉〈/math〉 is a real parameter and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mi〉g〈/mi〉〈/math〉 is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Yoshikazu Giga, Zhongyang Gu, Pen-Yuan Hsu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with a regularity criterion based on vorticity direction for Navier–Stokes equations in a three-dimensional bounded domain under the no-slip boundary condition. It asserts that if the vorticity direction is uniformly continuous in space uniformly in time, there is no type I blow-up. A similar result has been proved for a half space by Y. Maekawa and the first and the last authors (2014). The result of this paper is its natural but non-trivial extension based on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈/math〉 theory of the Stokes and the Navier–Stokes equations recently developed by K. Abe and the first author.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Monica Conti, Stefania Gatti, Alain Miranville〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Our aim in this paper is to study a mathematical model for the proliferative-to-invasive transition of hypoxic glioma cells. We prove the existence and uniqueness of nonnegative solutions and then address the important question of whether the positive solutions undergo extinction or permanence. More precisely, we prove that this depends on the boundary conditions: there is no extinction when considering Neumann boundary conditions, while we prove extinction when considering Dirichlet boundary conditions.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Meiqiang Feng, Xuemei Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Consider the existence, nonexistence and global estimates of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si26.svg"〉〈mi〉k〈/mi〉〈/math〉-convex solutions to the boundary blow-up 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si26.svg"〉〈mi〉k〈/mi〉〈/math〉-Hessian problem 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si3.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉H〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉[〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mrow〉〈mo〉[〈/mo〉〈mo〉ln〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈mtext〉for〈/mtext〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="0.2777em"〉〈/mspace〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉→〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉∞〈/mi〉〈mtext〉as〈/mtext〉〈mi mathvariant="normal"〉dist〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉→〈/mo〉〈mn〉0〈/mn〉〈mo〉.〈/mo〉〈/mrow〉〈/math〉 Here 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉k〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉{〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mo〉…〈/mo〉〈mo〉,〈/mo〉〈mi〉N〈/mi〉〈mo〉}〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mrow〉〈/math〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si26.svg"〉〈mi〉k〈/mi〉〈/math〉-Hessian operator, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mi〉Ω〈/mi〉〈/math〉 is a smooth, bounded, strictly convex domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mspace width="0.16667em"〉〈/mspace〉〈mrow〉〈mo〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉H〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is a positive weight function which is singular near the boundary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈/mrow〉〈/math〉. We first give the existence and nonexistence results of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si26.svg"〉〈mi〉k〈/mi〉〈/math〉-convex solution to the above boundary blow-up problem on a larger range of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mi〉H〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mi〉β〈/mi〉〈/math〉. Then we show that there is a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si26.svg"〉〈mi〉k〈/mi〉〈/math〉-convex solution provided that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉H〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 grows fast near 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si18.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mrow〉〈mo〉[〈/mo〉〈mo〉ln〈/mo〉〈mi〉u〈/mi〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 grows slow at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si19.svg"〉〈mi〉∞〈/mi〉〈/math〉. It turns out that this case is more difficult to handle than the case in which 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉H〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 grows slow near 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si18.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mrow〉〈mo〉[〈/mo〉〈mo〉ln〈/mo〉〈mi〉u〈/mi〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 grows fast at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si19.svg"〉〈mi〉∞〈/mi〉〈/math〉. This needs some new ingredients in the arguments.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Leonardo P.C. da Cruz, Valery G. Romanovski, J. Torregrosa〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study a family of quartic linear-like reversible polynomial systems having a nondegenerate center at the origin. This family has degree one with respect to one of the variables. We are interested in systems in this class having two extra nondegenerate centers outside the straight line of symmetry. The geometrical configuration of these centers is aligned or triangular. We solve the center problem in both situations and, in the second case, we study the limit cycles obtained from a simultaneous degenerate Hopf bifurcation in the quartic polynomials class.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Van Duong Dinh〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the existence and stability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction in dimensions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉d〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mn〉3〈/mn〉〈/mrow〉〈/math〉. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mi〉d〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉4〈/mn〉〈/mrow〉〈/math〉. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Zhuan Ye〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The purpose of this paper is to provide an alternative approach to the global regularity for the two-dimensional Euler–Boussinesq equations which couple the incompressible Euler equation for the velocity and a transport equation with fractional critical diffusion for the temperature. In contrast to the first proof of this result in [T. Hmidi, S. Keraani, and F. Rousset, Comm. Partial Differential Equations, 36 (2011), pp. 420–445] that took fully exploit of the hidden structure of the coupling system, the main argument in this manuscript is mainly based on the differentiability of the drift–diffusion equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Olimpio H. Miyagaki, Cláudia R. Santana, Rônei S. Vieira〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the existence of nontrivial ground state solutions for the following class of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian type equation 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mtable align="axis" equalrows="false" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉div〈/mi〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉a〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfenced〉〈mo〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉α〈/mi〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak"〉∗〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉α〈/mi〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak"〉∗〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉3〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mo〉−〈/mo〉〈mi〉∞〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mfrac〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉e〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉d〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉e〈/mi〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≔〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉e〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉N〈/mi〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mi〉d〈/mi〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉 (critical Hardy–Sobolev exponent); 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mi〉f〈/mi〉〈/math〉 has a quasicritical growth; V and K are nonnegative potentials; the function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mi〉a〈/mi〉〈/math〉 satisfies 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉a〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mspace width="0.16667em"〉〈/mspace〉〈mspace width="0.16667em"〉〈/mspace〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mspace width="0.16667em"〉〈/mspace〉〈mspace width="0.16667em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈msub〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 for any 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mrow〉〈mi〉ξ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, a.e. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, for any two positive functions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si15.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉l〈/mi〉〈mi〉o〈/mi〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si16.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mover accent="false"〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mo accent="true"〉¯〈/mo〉〈/mover〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mrow〉〈mover accent="false"〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mo accent="true"〉¯〈/mo〉〈/mover〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉α〈/mi〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Shuhui He, Glen Wheeler, Valentina-Mira Wheeler〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The paper is a mathematical investigation of a curvature flow model for embryonic epidermal wound healing proposed by Ravasio et al. (2015). Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity, showing that the singularity profile after continuous rescaling is that of a circle. One of the key new results we require is a maximal time estimate, which is also of independent interest.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Ky Ho, Inbo Sim〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide fundamental properties of the first eigenpair for fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emanating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems involving singular weights. These are new results, even for (fractional) Laplacian.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 2 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Nicola Garofalo, Giulio Tralli〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a class of second-order partial differential operators 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi mathvariant="script"〉A〈/mi〉〈/math〉 of Hörmander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the ’30s. We analyse some properties of the nonlocal operators driven by the fractional powers of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi mathvariant="script"〉A〈/mi〉〈/math〉, and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincaré inequalities adapted to the underlying geometry.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Shigeaki Koike, Andrzej Święch, Shota Tateyama〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The weak Harnack inequality for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-viscosity solutions is derived from the weak Harnack inequality for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-viscosity supersolutions. The local maximum principle for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-viscosity subsolutions and the Harnack inequality for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Luca Alasio, Maria Bruna, José Antonio Carrillo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that solutions of nonlinear nonlocal Fokker–Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si26.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Martin Dindoš, Luke Dyer〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉L〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mtext〉div〈/mtext〉〈mrow〉〈mo〉(〈/mo〉〈mi〉A〈/mi〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mo〉Lip〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 time-varying cylinders, where the coefficient matrix 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉A〈/mi〉〈mo〉=〈/mo〉〈mfenced open="[" close="]"〉〈mrow〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfenced〉〈/math〉 is uniformly elliptic and bounded.〈/p〉 〈p〉We show that if the Regularity problem 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈msub〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 for the equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉L〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 is solvable for some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mn〉1〈/mn〉〈mo〉〈〈/mo〉〈mi〉p〈/mi〉〈mo〉〈〈/mo〉〈mi〉∞〈/mi〉〈/math〉 then the Dirichlet problem 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈msub〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈/math〉 for the adjoint equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mi〉v〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 is also solvable, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈mi〉p〈/mi〉〈mo〉∕〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉. This result is an analogue of the result established in the elliptic case by Kenig and Pipher (1993). In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis (1996) and Nyström (2006) for scalar parabolic systems. In comparison, our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains.〈/p〉 〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Huatao Chen, Juan Luis García Guirao, Dengqing Cao, Jingfei Jiang, Xiaoming Fan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper concerns with the long time dynamical behavior of a stochastic Euler–Bernoulli beam driven by additive white noise. By verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam, the existence of global random attractors that attracts all bounded sets in phase space is proved. Furthermore, the finite Hausdorff dimension for the global random attractors is attained. In light of the relationship between global random attractor and random invariant probability measure, the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Marco Degiovanni, Marco Marzocchi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of “natural constraint”.〈/p〉〈/div〉

Description: 〈p〉Publication date: May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 182〈/p〉 〈p〉Author(s): Shangbing Ai, Craig Cowan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work we consider the existence of positive solutions to various equations of the form 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si1.gif"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none none none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mtd〉〈mtd columnalign="center"〉〈mo〉=〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mi〉g〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/mtd〉〈mtd columnalign="right"〉〈mtext〉in 〈/mtext〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈/mtd〉〈mtd columnalign="center"〉〈mo〉=〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mn〉0〈/mn〉〈/mtd〉〈mtd columnalign="right"〉〈mtext〉on 〈/mtext〉〈mi〉∂〈/mi〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msub〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is the open ball of radius 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉R〈/mi〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 centered at the origin and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉p〈/mi〉〈mo〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉N〈/mi〉〈mo〉+〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉. We will generally assume 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉g〈/mi〉〈/math〉 is nonnegative. Our approach will be to utilize some dynamical systems approaches.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Yao Lu, Yongqiang Fu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we deal with the multiplicity existence of solutions for the following 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mi〉p〈/mi〉〈/math〉-biharmonic equation: 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mspace width="2em"〉〈/mspace〉〈mspace width="2em"〉〈/mspace〉〈mspace width="2em"〉〈/mspace〉〈mtext〉on〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mi〉Ω〈/mi〉〈/math〉 is a bounded domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉Δ〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉N〈/mi〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈mi〉p〈/mi〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/math〉 is a parameter. When 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, we prove that the above problem possesses infinitely many solutions. While when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, a multiplicity existence result is obtained.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 191〈/p〉 〈p〉Author(s): P. Poláčik, P. Quittner〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In studies of superlinear parabolic equations 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉t〈/mi〉〈mo linebreak="goodbreak"〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mrow〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, backward self-similar solutions play an important role. These are solutions of the form 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉T〈/mi〉〈mo〉−〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/msup〉〈mi〉w〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉y〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≔〈/mo〉〈mi〉x〈/mi〉〈mo〉∕〈/mo〉〈msqrt〉〈mrow〉〈mi〉T〈/mi〉〈mo〉−〈/mo〉〈mi〉t〈/mi〉〈/mrow〉〈/msqrt〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mi〉T〈/mi〉〈/math〉 is a constant, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉w〈/mi〉〈/math〉 is a solution of the equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉Δ〈/mi〉〈mi〉w〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉y〈/mi〉〈mi〉⋅〈/mi〉〈mo〉∇〈/mo〉〈mi〉w〈/mi〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉w〈/mi〉〈mo〉∕〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈msup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉. We consider (classical) positive radial solutions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉w〈/mi〉〈/math〉 of this equation. Denoting by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈/msub〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉J〈/mi〉〈mi〉L〈/mi〉〈/mrow〉〈/msub〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈/msub〉〈/math〉 the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉J〈/mi〉〈mi〉L〈/mi〉〈/mrow〉〈/msub〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 there are only countably many solutions, and for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉J〈/mi〉〈mi〉L〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈/msub〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Volker Branding〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We derive the stress–energy tensor for polyharmonic maps between Riemannian manifolds. Moreover, we employ the stress–energy tensor to characterize polyharmonic maps where we pay special attention to triharmonic maps.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Teresa Isernia〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Using a minimization argument and a quantitative deformation lemma, we establish the existence of least energy sign-changing solutions for the following nonlinear Kirchhoff problem 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉a〈/mi〉〈mo〉+〈/mo〉〈mi〉b〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉[〈/mo〉〈mi〉u〈/mi〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈/mrow〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 are constants, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is the fractional Laplacian, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉V〈/mi〉〈mo〉,〈/mo〉〈mi〉K〈/mi〉〈/mrow〉〈/math〉 are continuous, positive functions, allowed for vanishing behavior at infinity, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉f〈/mi〉〈/math〉 is a continuous function satisfying suitable growth assumptions. Moreover, when the nonlinearity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉f〈/mi〉〈/math〉 is odd, we obtain the existence of infinitely many nontrivial weak solutions not necessarily nodals.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Vincenzo Ambrosio〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We complete the recent study started in Ambrosio (2019) concerning the existence and concentration phenomenon of complex-valued solutions for a class of nonlinear Schrödinger equations driven by the fractional magnetic Laplacian 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉. The proofs are obtained by combining suitable variational methods with a Kato’s approximation argument for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉. The approach developed here can be also used to consider other fractional magnetic problems like fractional magnetic Choquard equations, fractional magnetic Kirchhoff problems and fractional magnetic Schrödinger–Poisson equations, in which local conditions on the potential are assumed.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 191〈/p〉 〈p〉Author(s): Savin Treanţă, Manuel Arana-Jiménez, Tadeusz Antczak〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, optimality conditions are investigated for a class of PDE&PDI-constrained variational control problems. Thus, an efficient condition for a local optimal solution of the considered PDE&PDI-constrained variational control problem to be its global optimal solution is derived. The theoretical development is supported by a suitable example of nonconvex optimization problem.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Myong-Hwan Ri〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we prove that a Leray–Hopf weak solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉u〈/mi〉〈/math〉 to 3D Navier–Stokes equations is regular if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msubsup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mo〉̇〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉-norm of a suitable low frequency part of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉u〈/mi〉〈/math〉 is bounded by a scaling invariant constant depending on the kinematic viscosity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mi〉ν〈/mi〉〈/math〉 and initial value 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉. Moreover, we prove that a Leray–Hopf weak solution is regular if its medium frequency part with Fourier modes between 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉k〈/mi〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mi〉k〈/mi〉〈/math〉 for a sufficiently high wavenumber 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mi〉k〈/mi〉〈/math〉 has small 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈msubsup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mo〉̇〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈/math〉-norm. Our results imply that energy concentration at sufficiently high wavenumber bands bringing about singularity of the incompressible Navier–Stokes flow can be prevented by an “energy threshold” at a lower wavenumber band.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): C.A. Stuart〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In an open, bounded subset 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉Ω〈/mi〉〈/math〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi〉Ω〈/mi〉〈/mrow〉〈/math〉 we consider the nonlinear eigenvalue problem 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si4.svg"〉〈mrow〉〈mo〉−〈/mo〉〈munderover〉〈mrow〉〈mo linebreak="badbreak"〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉,〈/mo〉〈mi〉j〈/mi〉〈mo linebreak="badbreak"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉{〈/mo〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉}〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉n〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉g〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉u〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈msub〉〈mrow〉〈mo linebreak="newline" indentalign="id" indenttarget="mmlalignd1e194"〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈munderover〉〈mrow〉〈mo linebreak="badbreak"〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉,〈/mo〉〈mi〉j〈/mi〉〈mo linebreak="badbreak"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak"〉〈〈/mo〉〈mi〉∞〈/mi〉〈mspace width="1em"〉〈/mspace〉〈mtext〉and〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" indentalign="id" indenttarget="mmlalignd1e194"〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mtext〉on〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉V〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and the nonlinear terms 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mi〉n〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mi〉g〈/mi〉〈/math〉 are of higher order near 0 so that the formal linearization about the trivial solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≡〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si10.svg"〉〈mrow〉〈mo〉−〈/mo〉〈munderover〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉,〈/mo〉〈mi〉j〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉{〈/mo〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉}〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉V〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉u〈/mi〉〈mo〉.〈/mo〉〈/mrow〉〈/math〉 The leading term is degenerate elliptic on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉Ω〈/mi〉〈/math〉 because it is assumed that there are constants 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si13.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉|〈/mo〉〈mi〉ξ〈/mi〉〈msup〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈munderover〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉,〈/mo〉〈mi〉j〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉ξ〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉ξ〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈msup〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉|〈/mo〉〈mi〉ξ〈/mi〉〈msup〉〈mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mspace width="1em"〉〈/mspace〉〈mtext〉for all〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉ξ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mtext〉and almost all〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi〉Ω〈/mi〉〈mo〉.〈/mo〉〈/mrow〉〈/math〉 This is the lowest level of degeneracy at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 for which the linearization has a non-empty essential spectrum. Furthermore, elliptic regularity theory does not apply at 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉. Eigenfunctions of the linearization and solutions of the nonlinear problem having finite energy may be singular at the origin. The main results establish conditions for the existence or not of eigenvalues of the linearization, describe the behaviour of eigenfunctions as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si16.svg"〉〈mrow〉〈mi〉x〈/mi〉〈mo〉→〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 and determine values of the parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉λ〈/mi〉〈/math〉 at which bifurcation from the line of trivial solutions occurs. Standard bifurcation theory does not apply, even when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mi〉n〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mi〉g〈/mi〉〈/math〉 are smooth functions, since the nonlinear terms generate operators which are Gâteaux but not Fréchet differentiable at the trivial solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 191〈/p〉 〈p〉Author(s): Maya Chhetri, Petr Girg〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the following nonlocal problem 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none none none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="right"〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mtd〉〈mtd columnalign="center"〉〈mo〉=〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉λ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉λ〈/mi〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉;〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈mi〉u〈/mi〉〈/mtd〉〈mtd columnalign="center"〉〈mo〉=〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉∖〈/mo〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is the fractional Laplacian operator with fixed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉Ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉〉〈/mo〉〈mn〉2〈/mn〉〈mi〉s〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is a bounded domain with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 boundary, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 is a constant and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/math〉 is a bifurcation parameter. Here 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mo〉:〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉×〈/mo〉〈mi〉Ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉×〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉→〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/math〉 is a Carathéodory function that is sublinear at infinity. We use bifurcation theory to establish the existence of continua of the solution set bifurcating from infinity at the principal eigenvalue of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and discuss the nodal properties of solutions on these continua. We establish the multiplicity of solutions near the resonance and the existence of solution in the resonant case. As corollaries, we obtain anti-maximum principle and solvability for the resonant case satisfying the so called Landesman–Lazer type condition.〈/p〉〈/div〉

Description: 〈p〉Publication date: February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 191〈/p〉 〈p〉Author(s): Junyong Eom, Kazuhiro Ishige〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 be a solution to the Cauchy problem for a nonlinear parabolic system 〈span〉〈span〉(P)〈/span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/mtd〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mi〉v〈/mi〉〈mo〉=〈/mo〉〈mi〉Δ〈/mi〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mtd〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mo〉+〈/mo〉〈mi〉φ〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/mtd〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉v〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉μ〈/mi〉〈mo〉+〈/mo〉〈mi〉ψ〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/mtd〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mi〉α〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo〉,〈/mo〉〈mi〉μ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉φ〈/mi〉〈mo〉,〈/mo〉〈mi〉ψ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi〉B〈/mi〉〈mi〉C〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="0.16667em"〉〈/mspace〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∩〈/mo〉〈mspace width="0.16667em"〉〈/mspace〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉r〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉∞〈/mi〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mrow〉〈msub〉〈mrow〉〈mo〉inf〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mi〉φ〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉λ〈/mi〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈msub〉〈mrow〉〈mo〉inf〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mi〉ψ〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉μ〈/mi〉〈/mrow〉〈/math〉. Then the solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 to problem (P) behaves like a positive solution to ODE’s 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si16.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉ζ〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mrow〉〈msup〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mi〉ζ〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si18.svg"〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 and both of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si19.svg"〉〈msub〉〈mrow〉〈mo〉‖〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/msub〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si20.svg"〉〈msub〉〈mrow〉〈mo〉‖〈/mo〉〈mi〉v〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="bold"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/msub〉〈/math〉 diverge as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si21.svg"〉〈mrow〉〈mi〉t〈/mi〉〈mo〉→〈/mo〉〈mi〉∞〈/mi〉〈/mrow〉〈/math〉. In this paper we obtain the precise description of the large time behavior of the solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 24 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Alessandro Fonda, Giuliano Klun, Andrea Sfecci〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove the existence of periodic solutions of some infinite-dimensional nearly integrable Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of a generalization of the Poincaré–Birkhoff Theorem.〈/p〉〈/div〉

Description: 〈p〉Publication date: June 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 195〈/p〉 〈p〉Author(s): Xiaoyan Lin, Jiuyang Wei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the following singularly perturbed Kirchhoff-type problem 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mfenced open="(" close=")"〉〈mrow〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉a〈/mi〉〈mo〉+〈/mo〉〈mi〉ε〈/mi〉〈mi〉b〈/mi〉〈msub〉〈mrow〉〈mo linebreak="badbreak"〉∫〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi mathvariant="normal"〉d〈/mi〉〈mi〉x〈/mi〉〈/mrow〉〈/mfenced〉〈mo〉△〈/mo〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo〉;〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉ε〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 is a small parameter, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 are two constants, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉V〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is of super-linear growth at infinity and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor monotonicity condition on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∕〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉. By using some new techniques and subtle analyses, we prove that there exists a constant 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 determined by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mi〉V〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mi〉f〈/mi〉〈/math〉 such that for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈mi〉ε〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉]〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, the above problem has a ground state solution concentrating around global minimum of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mi〉V〈/mi〉〈/math〉 in the semi-classical limit. Our results are available to the case that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∼〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mrow〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si15.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mn〉6〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, and extend the existing results concerning the case that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∼〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mrow〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉[〈/mo〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mn〉6〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: March 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 192〈/p〉 〈p〉Author(s): Konrad Kisiel, Krzysztof Chełmiński〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this article we study the existence theory to the Prandtl–Reuss dynamical model of elasto-perfect plasticity with non-homogeneous mixed boundary conditions. By using only the Yosida approximation of inelastic constitutive multifunction we are able to prove the existence of solutions without assuming any kind of safe-load conditions, which are quite common in the theory of elasto-perfect plasticity.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Mark Allen, Mariana Smit Vega Garcia〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉χ〈/mi〉〈/mrow〉〈mrow〉〈mrow〉〈mo〉{〈/mo〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈mi〉c〈/mi〉〈mo〉}〈/mo〉〈/mrow〉〈/mrow〉〈/msub〉〈/math〉〈/span〉for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉s〈/mi〉〈mo〉〈〈/mo〉〈mn〉1〈/mn〉〈/math〉 and an arbitrary constant 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉c〈/mi〉〈/math〉. Our main object of study is the free boundary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉∂〈/mi〉〈mrow〉〈mo〉{〈/mo〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈mi〉c〈/mi〉〈mo〉}〈/mo〉〈/mrow〉〈/math〉. We study the behavior of the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉s〈/mi〉〈mo〉≤〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/math〉 certain symmetric solutions are stable; however, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉s〈/mi〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/math〉 these solutions are not stable and therefore not minimizers of the corresponding functional.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 9 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Nestor Guillen, Russell W. Schwab〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min–max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min–max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Sujin Khomrutai〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We investigate a nonlocal equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mi〉J〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉−〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉y〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉d〈/mi〉〈mi〉y〈/mi〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mo〉‖〈/mo〉〈mi〉J〈/mi〉〈mo〉‖〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉+〈/mo〉〈mi〉a〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉a〈/mi〉〈/math〉 is unbounded and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉J〈/mi〉〈/math〉 belongs to a weighted space. Crucial weighted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si15.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and interpolation estimates for the Green operator are established by a new method based on the sharp Young’s inequality, the asymptotic behavior of a regular varying coefficients exponential series, and the properties of auxiliary functions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉Γ〈/mi〉〈mo〉=〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∕〈/mo〉〈mi〉η〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉b〈/mi〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈mo〉−〈/mo〉〈mi〉Γ〈/mi〉〈mo〉∕〈/mo〉〈mi〉η〈/mi〉〈mo〉≲〈/mo〉〈mi〉J〈/mi〉〈mo〉∗〈/mo〉〈mi〉Γ〈/mi〉〈mo〉−〈/mo〉〈mi〉Γ〈/mi〉〈mo〉≲〈/mo〉〈mi〉Γ〈/mi〉〈mo〉∕〈/mo〉〈mi〉η〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈msup〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mo〉+〈/mo〉〈/mrow〉〈/msub〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉≲〈/mo〉〈mi〉Γ〈/mi〉〈mo〉∕〈/mo〉〈msup〉〈mrow〉〈mfenced open="〈" close="〉"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/mfenced〉〈/mrow〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈/msup〉〈mo〉≲〈/mo〉〈msup〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈/mrow〉〈/msub〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉. Blow-up behaviors are investigated by employing test functions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈msub〉〈mrow〉〈mi〉ϕ〈/mi〉〈/mrow〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mi〉Γ〈/mi〉〈/math〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si10.gif"〉〈mi〉η〈/mi〉〈mo〉=〈/mo〉〈mi〉R〈/mi〉〈/math〉) instead of principal eigenfunctions. Global well-posedness in weighted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si15.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 spaces for the Cauchy problem is proved. When 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si12.gif"〉〈mi〉a〈/mi〉〈mo〉∼〈/mo〉〈msup〉〈mrow〉〈mfenced open="〈" close="〉"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/mfenced〉〈/mrow〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 the Fujita exponent is shown to be 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si13.gif"〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉σ〈/mi〉〈mo〉+〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∕〈/mo〉〈mi〉n〈/mi〉〈/math〉. Our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Setenay Akduman, Alexander Pankov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The paper deals with nonlinear Schrödinger equations on infinite metric graphs. We assume that the linear potential is infinitely growing. We prove an existence and multiplicity result that covers both self-focusing and defocusing cases. Furthermore, under some additional assumptions we show that solutions obtained bifurcate from trivial ones. We prove that these solutions are superexponentially localized. Our approach is variational and based on generalized Nehari manifold.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Daisuke Naimen, Masataka Shibata〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We investigate the Kirchhoff type elliptic problem with critical nonlinearity; 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si1.gif"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mfenced open="(" close=")"〉〈mrow〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mi〉α〈/mi〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈/mrow〉〈/mfenced〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈mtext〉on〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉5〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉Ω〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is a bounded domain with smooth boundary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉α〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉λ〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈mn〉2〈/mn〉〈mi〉N〈/mi〉〈mo〉∕〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉N〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈mi〉q〈/mi〉〈mo〉∈〈/mo〉〈mrow〉〈mo〉[〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msup〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉. We prove the existence of two solutions of it via the variational method. Since 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉5〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉α〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, the uniqueness assertion for the associated limiting problem may fail. This causes serious difficulties in controlling concentrating Palais–Smale sequences. We overcome these by introducing new techniques. For a mountain pass type solution, we utilize the limit function of the fibering maps of the concentrating Palais–Smale sequence. This tool is based on our careful setting of Nehari type sets. On the other hand, a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution. This is the first work which proves the multiplicity of positive solutions of the Kirchhoff type critical problem in high dimension.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Wei Lian, Md Salik Ahmed, Runzhang Xu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider the semilinear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with logarithmic Sobolev inequality, we derive the global existence and infinite time blow up of the solution at low energy level 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉E〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉〈〈/mo〉〈mi〉d〈/mi〉〈/math〉 . Then these results are extended in parallel to the critical case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si59.gif"〉〈mi〉E〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉d〈/mi〉〈/math〉. Besides, with additional assumptions on initial data, the infinite time blow up result is given with arbitrary positive initial energy 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si62.gif"〉〈mi〉E〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Regularity properties are investigated for the value function of the Bolza optimal control problem with affine dynamic and end-point constraints. In the absence of singular geodesics, we prove the local semiconcavity of the sub-Riemannian distance from a compact set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉Γ〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉. Such a regularity result was obtained by the second author and L. Rifford in Cannarsa and Rifford (2008) when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉Γ〈/mi〉〈/math〉 is a singleton. Furthermore, we derive sensitivity relations for time optimal control problems with general target sets 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉Γ〈/mi〉〈/math〉, that is, without imposing any geometric assumptions on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉Γ〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Jiecheng Chen, Dashan Fan, Fayou Zhao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We establish Littlewood–Paley characterizations of the Sobolev spaces 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉 in Euclidean spaces using several square functions defined via the spherical average, the ball average, the Bochner–Riesz means and some other well known operators. We provide a simple proof so that we are able to extend and improve many results published in recent papers.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 8 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Paolo Antonelli, Pierangelo Marcati, Hao Zheng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with an existence and stability result on the nonlinear derivative Schrödinger equation in 1-D, which is originated by the study of the stability of nontrivial steady states in Quantum Hydrodynamics. The problem is equivalent to a compressible Euler fluid system with a very specific Korteweg–Kirchhoff stress 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉ρ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mfrac〉〈mrow〉〈mo〉ħ〈/mo〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mi〉ρ〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉. As a simple, but significative, example we consider the nonlinear derivative Schrödinger equation obtained via a complex Cole–Hopf type transformation, applied to the 1-D free Schrödinger equation. The resulting problem (possibly unstable) is investigated for small solutions around the null steady state. The stability is proved to be valid for long time intervals of order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉O〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉4〈/mn〉〈mo〉∕〈/mo〉〈mn〉5〈/mn〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉ϵ〈/mi〉〈/math〉 is the order of smallness of the initial data. This result brought back to the QHD system provides the stability of the steady state 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉ρ〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉J〈/mi〉〈mo〉=〈/mo〉〈mi〉v〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉. The validity in time of this result is far beyond what can be obtained via classical linearization analysis or via higher order energy estimates. Indeed in our analysis the nonlinear structure plays a crucial role in the corresponding iteration procedure, the use of local smoothing and the Schrödinger maximal operator provides the control of the potential lost of regularity.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 19 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Pietro Celada, Jihoon Ok〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study partial 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉  – regularity of minimizers of quasi-convex variational integrals with non-standard growth. We assume in particular that the relevant integrands satisfy an Orlicz’s type growth condition, i.e. a so-called general growth condition. Moreover, the functionals are supposed to be non-autonomous and possibly degenerate.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Wei Ding, Guozhen Lu, YuePing Zhu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Though multi-parameter Hardy space theory has been well developed in the past half century, not much has been studied for a local Hardy space theory in the multi-parameter settings. Such multi-parameter local Hardy spaces can play an important role in studying the boundedness of multi-parameter pseudo-differential operators, multi-parameter singular integrals of non-convolution type, and applications to partial differential equations, etc. By establishing a bi-parameter local reproducing formula, bi-parameter local Hardy space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 is introduced in this paper. This space coincides with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉p〈/mi〉〈mo〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉. While 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉p〈/mi〉〈mo〉≤〈/mo〉〈mn〉1〈/mn〉〈/math〉, this space is substantially different from the classical bi-parameter Hardy spaces 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉. We will establish its atomic decomposition of the bi-parameter local Hardy spaces and as an application, we prove the boundedness from 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 of the bi-parameter singular integral operators in inhomogeneous Journé class for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈mi〉p〈/mi〉〈/math〉 near 1. For the simplicity, we have chosen to present all the results in the bi-parameter setting. Nevertheless, all of them hold for arbitrary number of parameters. The multi-parameter local theory developed in this paper can serve as a model case for similar theory in other multi-parameter settings.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Phuong Le〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉α〈/mi〉〈mo〉〈〈/mo〉〈mn〉2〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉β〈/mi〉〈mo〉〈〈/mo〉〈mi〉n〈/mi〉〈/math〉. We prove that the equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si4.gif"〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mfenced open="(" close=")"〉〈mrow〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo〉∗〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfenced〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 has no positive solution if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mn〉1〈/mn〉〈mo〉≤〈/mo〉〈mi〉p〈/mi〉〈mo〉〈〈/mo〉〈mfrac〉〈mrow〉〈mi〉n〈/mi〉〈mo〉+〈/mo〉〈mi〉β〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉. We also classify all positive solutions to the equation in the critical case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉p〈/mi〉〈mo〉=〈/mo〉〈mfrac〉〈mrow〉〈mi〉n〈/mi〉〈mo〉+〈/mo〉〈mi〉β〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 19 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Francescantonio Oliva, Francesco Petitta〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si1.gif"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉h〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉f〈/mi〉〈mo〉+〈/mo〉〈mi〉μ〈/mi〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉on〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉{〈/mo〉〈mn〉0〈/mn〉〈mo〉}〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉Ω〈/mi〉〈/math〉 is an open bounded subset of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉N〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉), 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a nonnegative integrable function, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈mi〉p〈/mi〉〈/math〉-Laplace operator, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈mi〉μ〈/mi〉〈/math〉 is a nonnegative bounded Radon measure on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si10.gif"〉〈mi〉f〈/mi〉〈/math〉 is a nonnegative function of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si11.gif"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉. The term 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si12.gif"〉〈mi〉h〈/mi〉〈/math〉 is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si12.gif"〉〈mi〉h〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Brian Allen, Edward Bryden〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the Hölder sense. This is done by establishing a general trace inequality on Riemannian manifolds which is an interesting result on its own. We provide examples demonstrating how each of our hypotheses are necessary. In the Appendix by the first author with Christina Sormani, we prove that sequences of compact integral current spaces without boundary (including Riemannian manifolds) that have uniform Hölder bounds on their distance functions have subsequences converging in the Gromov–Hausdorff (GH) sense. If in addition they have a uniform upper bound on mass (volume) then they converge in the Sormani–Wenger Intrinsic Flat (SWIF) sense to a limit whose metric completion is the GH limit. We provide an example of a sequence developing a cusp demonstrating why the SWIF and GH limits may not agree.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Claudio Arancibia-Ibarra〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉I analyse a modified May–Holling–Tanner predator–prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator–prey population. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle–node, Hopf, and Bogdanov–Takens bifurcations. We use simulations to illustrate the behaviour of the model.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Sungchol Kim, Dukman Ri〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the integral functionals with the general nonstandard growth. We prove the boundedness and Hölder continuity of quasiminimizers of these functionals. Our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in “Direct Methods in the Calculus of Variations, 2003” by Giusti.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Irena Lasiecka, Michael Pokojovy, Xiang Wan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider an initial–boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell–Cattaneo–Vernotte law giving rise to a ‘second sound’ effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/math〉-spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small – not in the full topology of our solution class, but in a lower topology corresponding to weak solutions we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Marino Badiale, Stefano Greco, Sergio Rolando〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given three measurable functions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉V〈/mi〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/mfenced〉〈mo〉≥〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉K〈/mi〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/mfenced〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉Q〈/mi〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/mfenced〉〈mo〉≥〈/mo〉〈mn〉0〈/mn〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉r〈/mi〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉, we consider the bilaplacian equation 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si5.gif"〉〈msup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉K〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉+〈/mo〉〈mi〉Q〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉〈/span〉and we find radial solutions thanks to compact embeddings of radial spaces of Sobolev functions into sum of weighted Lebesgue spaces.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Sitong Chen, Binlin Zhang, Xianhua Tang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the following singularly perturbed problem in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si17.gif"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si2.gif"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉V〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈munderover〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/munderover〉〈msubsup〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉ε〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉ε〈/mi〉〈mi〉Δ〈/mi〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉ε〈/mi〉〈/math〉 is a small parameter, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉V〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉f〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉. By using some new variational and analytic techniques joined with the manifold of Pohoz̆aev–Nehari type, we prove that there exists a constant 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈msub〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 determined by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈mi〉V〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈mi〉f〈/mi〉〈/math〉 such that for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈mi〉ε〈/mi〉〈mo〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉]〈/mo〉〈/mrow〉〈/math〉, the above problem admits a semiclassical ground state solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si10.gif"〉〈msub〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mo〉ˆ〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈/math〉 with exponential decay at infinity. We also establish a new concentration behaviour of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si11.gif"〉〈mrow〉〈mo〉{〈/mo〉〈msub〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mo〉ˆ〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈mo〉}〈/mo〉〈/mrow〉〈/math〉 as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si12.gif"〉〈mi〉ε〈/mi〉〈mo〉→〈/mo〉〈mn〉0〈/mn〉〈/math〉. In particular, our results are available to the nonlinearity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si13.gif"〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∼〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si14.gif"〉〈mi〉s〈/mi〉〈mo〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mn〉6〈/mn〉〈mo〉]〈/mo〉〈/mrow〉〈/math〉, which extend the existing results concerning the case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si13.gif"〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∼〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si16.gif"〉〈mi〉s〈/mi〉〈mo〉〉〈/mo〉〈mn〉6〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Bin Ge, De-Jing Lv, Jian-Fang Lu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the present paper, in view of the variational approach, we consider the existence and multiplicity of weak solutions for a class of the double phase problem 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none none none none none none none none" equalcolumns="false" columnspacing="0.27em"〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi mathvariant="normal"〉div〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉a〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∇〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="0.2777em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈mtd columnalign="right"〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈mtd columnalign="right"〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉on〈/mtext〉〈mspace width="0.2777em"〉〈/mspace〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉. Firstly, by the Fountain and Dual Theorem with Cerami condition, we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉f〈/mi〉〈/math〉. Secondly, we prove that this problem has at least one nontrivial solution for any parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 small enough, and also that the solution blows up, in the Sobolev norm, as 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo〉→〈/mo〉〈msup〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mo〉+〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉. Finally, by imposing additional assumptions on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉f〈/mi〉〈/math〉, we establish the existence of infinitely many solutions by using Krasnoselskii’s genus theory for the above equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): G. Cardone, C. Perugia, C. Timofte〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction–diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying several metabolic processes taking place in living cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes. The coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms. These nonlinearities, which model, at the microscopic scale, various volume or surface reaction processes, give rise in the macroscopic model to different effects, such as the appearance of additional source or sink terms or of a non-standard diffusion matrix.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 4 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Miroslav Bulíček, Piotr Gwiazda, Martin Kalousek, Agnieszka Świerczewska-Gwiazda〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi mathvariant="script"〉N〈/mi〉〈/math〉-function. First, an existence result is shown under the assumption that the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi mathvariant="script"〉N〈/mi〉〈/math〉-function or its convex conjugate satisfies 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉-condition. The second result concerns the homogenization process for families of strongly nonlinear elliptic problems with the homogeneous Dirichlet boundary condition under above stated conditions on the elliptic operator, which is additionally assumed to be periodic in the spatial variable.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 3 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Petru Mironescu, Emmanuel Russ, Yannick Sire〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉Ω〈/mi〉〈/math〉 be a smooth bounded (simply connected) domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉u〈/mi〉〈/math〉 be a complex-valued measurable function on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉Ω〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/math〉 a.e. Assume that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈mi〉u〈/mi〉〈/math〉 belongs to a Besov space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈msubsup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉;〈/mo〉〈mi〉ℂ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉. We investigate whether there exists a real-valued function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈mi〉φ〈/mi〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉B〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉;〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈mi〉ı〈/mi〉〈mi〉φ〈/mi〉〈/mrow〉〈/msup〉〈/math〉. This complements the corresponding study in Sobolev spaces due to Bourgain, Brezis and the first author. The microscopic parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si10.gif"〉〈mi〉q〈/mi〉〈/math〉 turns out to play an important role in some limiting situations. The analysis of this lifting problem relies on some interesting new properties of Besov spaces, in particular a non-restriction property when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si11.gif"〉〈mi〉q〈/mi〉〈mo〉〉〈/mo〉〈mi〉p〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Zhigang Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the Cauchy problem of the chemotaxis-shallow water system with degenerate viscosity coefficients. A local existence of the unique regular solution is established when the initial data are arbitrarily large and include vacuum. Moreover, a Beal–Kate–Majda type blow-up criterion is provided.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Wenjing Chen, Yuyan Gui〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is devoted to study the existence of multiple solutions for the following fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si19.svg"〉〈mi〉p〈/mi〉〈/math〉-Kirchhoff problem 〈span〉〈span〉(0.1)〈/span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mi〉M〈/mi〉〈mfenced open="(" close=")"〉〈mrow〉〈msub〉〈mrow〉〈mo linebreak="badbreak"〉∫〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉y〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo linebreak="badbreak"〉+〈/mo〉〈mi〉p〈/mi〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈mi〉d〈/mi〉〈mi〉y〈/mi〉〈/mrow〉〈/mfenced〉〈msubsup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo linebreak="badbreak"〉−〈/mo〉〈mo〉△〈/mo〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mspace width="1em"〉〈/mspace〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mtext〉in〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉∖〈/mo〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈msubsup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mo〉△〈/mo〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉 denotes the fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si19.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplace operator, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mi〉Ω〈/mi〉〈/math〉 is a smooth bounded set in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 containing 0 with Lipschitz boundary, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mi〉M〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉a〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mi〉b〈/mi〉〈msup〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉a〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈mi〉b〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mrow〉〈mi〉θ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si15.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉θ〈/mi〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉r〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈msubsup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msubsup〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si16.svg"〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msubsup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mfrac〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉α〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mi〉p〈/mi〉〈mi〉s〈/mi〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/math〉 is the fractional critical Hardy–Sobolev exponent for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉n〈/mi〉〈/mrow〉〈/math〉. By using fibering maps and Nehari manifold, we obtain that the existence of multiple solutions to problem (0.1) for both Hardy–Sobolev subcritical and critical cases. In particular, the concentration compactness principle will be used to overcome the lack of compactness for the critical case.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Leandro M. Del Pezzo, Alexander Quaas〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈/math〉Laplacian with weight in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈mo〉.〈/mo〉〈/math〉 We also show a nonexistence result when the weight has positive integral.〈/p〉 〈p〉In addition, we show some qualitative properties of the first eigenfunction including a sharp decay estimate. Finally, we extend the decay result to the positive solutions of a Schrödinger type equation.〈/p〉 〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): M. Delgado, M. Molina-Becerra, J.R. Santos Júnior, A. Suárez〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A logistic equation in the whole space is considered. In this problem, a non-local perturbation is included. We establish a new sub–supersolution method for general nonlocal elliptic equations and, consequently, we obtain the existence of positive solutions of a nonlocal logistic equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Chang-Yu Guo, Chang-Lin Xiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉M〈/mi〉〈/math〉 be a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-smooth Riemannian manifold with boundary and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉 a complete 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-smooth Riemannian manifold. We show that each stationary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉p〈/mi〉〈/math〉-harmonic mapping 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉u〈/mi〉〈mo〉:〈/mo〉〈mi〉M〈/mi〉〈mo〉→〈/mo〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉, whose image lies in a compact subset of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉, is locally 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉 for some 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉α〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, provided that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉 is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉p〈/mi〉〈/math〉-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉M〈/mi〉〈/math〉 has non-negative Ricci curvature and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉N〈/mi〉〈/math〉 is simply connected with non-positive sectional curvature, we deduce a gradient estimate for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉-smooth weakly 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si17.svg"〉〈mi〉p〈/mi〉〈/math〉-harmonic mappings from which follows a Liouville-type theorem in the same setting.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Riccardo Durastanti〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study existence and regularity of weak solutions for the following 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si15.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian system 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉A〈/mi〉〈msup〉〈mrow〉〈mi〉φ〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈mo linebreak="badbreak"〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈mo linebreak="badbreak"〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mi〉φ〈/mi〉〈mo〉=〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉r〈/mi〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉φ〈/mi〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉φ〈/mi〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mi〉Ω〈/mi〉〈/math〉 is an open bounded subset of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≥〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mi〉v〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≔〈/mo〉〈mtext〉div〈/mtext〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mo〉∇〈/mo〉〈mi〉v〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∇〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si15.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian operator, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉A〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mrow〉〈mi〉r〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si13.svg"〉〈mrow〉〈mn〉0〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉θ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mi〉f〈/mi〉〈/math〉 belongs to a suitable Lebesgue space. In particular, we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Gleydson C. Ricarte, Janielly G. Araújo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove interior and up to boundary Lipschitz regularity of the viscosity solutions to a singular perturbation problem for a reaction–diffusion equation related to the normalized 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian equation.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Marta Menci, Marco Papi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we propose local and global existence results for the solution of systems characterized by the coupling of ODEs and PDEs. The coexistence of distinct mathematical formalisms represents the main feature of 〈em〉hybrid〈/em〉 approaches, in which the dynamics of interacting agents are driven by second-order ODEs, while reaction–diffusion equations are used to model the time evolution of a signal influencing them. We first present an existence result of the solution, locally in time. In particular, we generalize the framework of recent works, presented in the literature with a modeling and numerical approach, which have not been investigated from an analytical point of view so far. Then, the previous result is extended in order to obtain a global solution.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Xu Dong, Yuanhong Wei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is concerned with the problem of elliptic equations in annular domains involving derivative terms. Using nonlinear analysis methods, some results regarding existence of solutions are established. More precisely, at least one radial solution is obtained basing on Schauder’s fixed point theorem and contraction mapping theorem, respectively. Some iterative technique is also applied, which allows us to overcome difficulties that the nonlinearity is related to derivative terms.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Paolo Antonelli, Stefano Spirito〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider the Navier–Stokes–Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. Incontrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the 〈em〉a priori〈/em〉 bounds obtained by the energy and the BD entropy.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): Xueke Pu, Min Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper concerns the existence of global smooth solutions to the initial boundary value problem for a three-dimensional compressible quantum hydrodynamic model with damping and heat diffusion in a bounded domain in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉. Based on the continuation argument and the uniform 〈em〉a priori〈/em〉 estimates with respect to the time, we obtain the existence of global solutions in a bounded smooth domain provided that the initial perturbation around a constant state is small enough. The key difficulty is to deal with the higher order quantum terms, which do play an essential role in establishing the 〈em〉a priori〈/em〉 estimates. The boundary conditions finally adopted are the insulating boundary conditions.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Elkin Cárdenas, Willy Sierra〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given a compact manifold with boundary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉M〈/mi〉〈/math〉 of dimension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉m〈/mi〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈/math〉 and a nondegenerate Riemannian metric 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msub〉〈/math〉 having null scalar curvature, constant mean curvature, and unitary volume on the boundary, we show that the set of Riemannian metrics with null scalar curvature, constant mean curvature, and unitary volume on the boundary, near to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msub〉〈/math〉 is an embedded submanifold of the manifold of all Riemannian metrics on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉M〈/mi〉〈/math〉. Additionally, such submanifold is strongly transversal to the conformal classes. We also prove, using recent results of compactness, that conformal classes of metrics closed to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mo〉∗〈/mo〉〈/mrow〉〈/msub〉〈/math〉 contain only one of these metrics.〈/p〉〈/div〉

Description: 〈p〉Publication date: July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 184〈/p〉 〈p〉Author(s): E. Lokharu, E. Wahlén〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider steady three-dimensional gravity–capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Pengyan Wang, Pengcheng Niu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain (bounded or unbounded) in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 via using a direct method of moving planes. This extends the results in Wang and Niu (2017) and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Amy Peterson, Ambar N. Sengupta〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to Hermite polynomials in the large-N limit. We determine the behavior of the spherical Laplacian as well as zonal harmonic polynomials in the large-N limit.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Fei Yang, Liangdi Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mrow〉〈mo〉(〈/mo〉〈msup〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mi〉g〈/mi〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉e〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mi〉ϕ〈/mi〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 be a smooth metric measure space. In this paper, we derive a series of gradient estimates and a Harnack inequality for positive solutions of a nonlinear parabolic partial differential equation 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si2.gif"〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϕ〈/mi〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉q〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉a〈/mi〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉ln〈/mo〉〈mi〉u〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈/math〉〈/span〉in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msup〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉∞〈/mi〉〈mo〉,〈/mo〉〈mo〉+〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si4.gif"〉〈mi〉q〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo fence="true"〉(〈/mo〉〈mrow〉〈msup〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉∞〈/mi〉〈mo〉,〈/mo〉〈mo〉+〈/mo〉〈mi〉∞〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mo fence="true"〉)〈/mo〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉α〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 19 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Patrizia Pucci, Vicenţiu D. Rădulescu〈/p〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Zhaoxia Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Asymptotic behavior of the higher order spacial derivatives of the strong solution to the incompressible non-stationary magneto-hydrodynamic (MHD) equations is given in a half-space, which is a long-time difficult question. The main tools employed in this article are the non-stationary Stokes solution formula, and some 〈em〉a priori〈/em〉 estimates of the steady Stokes system in the half-space.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Tong Tang, Hongjun Gao, Qingkun Xiao〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study a viscous capillary model of plasma as a so called Navier–Stokes–Poisson–Korteweg model. Our purpose is to prove the existence of global weak solutions for large data in a two-dimensional torus. We utilize the effective velocity and some interesting identities to remove the restrictions on the coefficients. It is worth pointing out that we prove the critical case that the value of viscosity coefficient is equivalent to the capillary coefficient 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉ν〈/mi〉〈mo〉=〈/mo〉〈mi〉κ〈/mi〉〈/math〉. In this case, the B-D entropy and method used in Antonelli and Spirito (2017) cannot be applied directly. Moreover, there is no friction term and cold pressure term in the model. In some senses, we improve the previous results.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): Marcone C. Pereira, Julio D. Rossi, Nicolas Saintier〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈mi〉U〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈mi〉V〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈/math〉, we show that the solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si3.gif"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈/math〉 to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si4.gif"〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉+〈/mo〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msubsup〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="2em"〉〈/mspace〉〈mtext〉in〈/mtext〉〈mi〉U〈/mi〉〈mo〉×〈/mo〉〈mi〉ε〈/mi〉〈mi〉V〈/mi〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si5.gif"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉x〈/mi〉〈mo〉⁄〈/mo〉〈mo〉∈〈/mo〉〈mi〉U〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si7.gif"〉〈mi〉y〈/mi〉〈mo〉∈〈/mo〉〈mi〉ε〈/mi〉〈mi〉V〈/mi〉〈/math〉, verifies that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si8.gif"〉〈msub〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo〉̃〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉≔〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉ε〈/mi〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉→〈/mo〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si9.gif"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and estimate the rate of convergence in the uniform norm. Here 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si10.gif"〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si11.gif"〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈/math〉 are the fractional Laplacian in the 1st variable 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si12.gif"〉〈mi〉x〈/mi〉〈/math〉 (with a Dirichlet condition, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si13.gif"〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈/math〉 if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si6.gif"〉〈mi〉x〈/mi〉〈mo〉⁄〈/mo〉〈mo〉∈〈/mo〉〈mi〉U〈/mi〉〈/math〉) and in the 2nd variable 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si15.gif"〉〈mi〉y〈/mi〉〈/math〉 (with a Neumann condition, integrating only inside 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si16.gif"〉〈mi〉V〈/mi〉〈/math〉), respectively, that is, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si17.gif"〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈mfrac〉〈mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉w〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉−〈/mo〉〈mi〉w〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉+〈/mo〉〈mn〉2〈/mn〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉w〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll" altimg="si18.gif"〉〈msubsup〉〈mrow〉〈mi〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msubsup〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈/msub〉〈mfrac〉〈mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉z〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉y〈/mi〉〈mo〉−〈/mo〉〈mi〉z〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈mo〉+〈/mo〉〈mn〉2〈/mn〉〈mi〉t〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉z〈/mi〉〈mo〉.〈/mo〉〈/math〉〈/p〉〈/div〉

Description: 〈p〉Publication date: June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 183〈/p〉 〈p〉Author(s): Yonggang Chen, Baiyu Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Chen and Li (2018) developed maximum principles for the fractional p-Laplacian, which carry out the direct method of moving planes to the fractional p-Laplacian equation. In this paper, we generalize their method to the system case and obtain a symmetry result for the fractional p-Laplacian system on the whole space. We also consider the system on the upper half space and obtain a non-existence result.〈/p〉〈/div〉

Description: 〈p〉Publication date: August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉 〈p〉Author(s): Verena Bögelein, Pekka Lehtelä, Stefan Sturm〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We investigate the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msubsup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mi〉m〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msubsup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Raúl Ferreira〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the blow-up phenomena for the system 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈/msub〉〈mi〉J〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉−〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉y〈/mi〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∈〈/mo〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈/msub〉〈mi〉J〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉−〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉v〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉y〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉y〈/mi〉〈mo〉−〈/mo〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈msup〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈mspace width="1em"〉〈/mspace〉〈/mtd〉〈mtd columnalign="left"〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉∈〈/mo〉〈mi〉Ω〈/mi〉〈mo〉×〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉 Under certain hypothesis we prove that the solution blows up infinite time and we classify in terms of the parameters when both components blow up together (simultaneous blow-up), or when only one component blows up (non-simultaneous blow-up). We also prove the blow-up rate for a particular case.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Francesca Anceschi, Sergio Polidoro, Maria Alessandra Ragusa〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉loc〈/mi〉〈/mrow〉〈mrow〉〈mi〉∞〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉 estimates for positive solutions to the following degenerate second order partial differential equation of Kolmogorov type with measurable coefficients of the form 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mrow〉〈munderover〉〈mrow〉〈mo linebreak="badbreak"〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉,〈/mo〉〈mi〉j〈/mi〉〈mo linebreak="badbreak"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mfenced open="(" close=")"〉〈mrow〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfenced〉〈mo linebreak="goodbreak"〉+〈/mo〉〈munderover〉〈mrow〉〈mo linebreak="badbreak"〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉,〈/mo〉〈mi〉j〈/mi〉〈mo linebreak="badbreak"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉−〈/mo〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mspace width="1em" indentalign="id" indenttarget="mmlalignd1e172"〉〈/mspace〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mspace width="0.16667em"〉〈/mspace〉〈munderover〉〈mrow〉〈mo linebreak="badbreak"〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo linebreak="badbreak"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉−〈/mo〉〈munderover〉〈mrow〉〈mo linebreak="badbreak"〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo linebreak="badbreak"〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉∂〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mfenced open="(" close=")"〉〈mrow〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfenced〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉c〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mo〉…〈/mo〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉z〈/mi〉〈/mrow〉〈/math〉 is a point of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈msub〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 is a uniformly positive symmetric matrix with bounded measurable coefficients, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉b〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 is a constant matrix. We apply the Moser’s iteration method to prove the local boundedness of the solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mi〉u〈/mi〉〈/math〉 under minimal integrability assumption on the coefficients.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Micol Amar, Ida De Bonis, Giuseppe Riey〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove existence and homogenization results for a family of elliptic problems involving interfaces and a singular lower order term. These problems model heat or electrical conduction in composite media.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Zhisu Liu, Haijun Luo, Zhitao Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study Dancer–Fuc̆ik spectrum of the fractional Schrödinger operators which is defined as the set of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉β〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 such that 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si2.svg"〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo linebreak="badbreak"〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉+〈/mo〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak"〉=〈/mo〉〈mi〉α〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak"〉+〈/mo〉〈mi〉β〈/mi〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak"〉−〈/mo〉〈/mrow〉〈/msup〉〈mspace width="1em"〉〈/mspace〉〈mtext〉in 〈/mtext〉〈mspace width="0.16667em"〉〈/mspace〉〈mspace width="0.16667em"〉〈/mspace〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉〈/span〉has a nontrivial solution 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉u〈/mi〉〈/math〉, where the potential 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈/math〉 has a steep potential well for sufficiently large parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉λ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉. It is allowed that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/math〉 has essential spectrum with finitely many eigenvalues below the infimum of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈mrow〉〈mi〉e〈/mi〉〈mi〉s〈/mi〉〈mi〉s〈/mi〉〈/mrow〉〈/msub〉〈mfenced open="(" close=")"〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msup〉〈mo〉+〈/mo〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉λ〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/mfenced〉〈/mrow〉〈/math〉. Many difficulties are caused by general nonlocal operators, we develop new techniques to overcome them to construct the first nontrivial curve of Dancer–Fuc̆ik point spectrum by minimax methods, to show some qualitative properties of the curve, and to prove that the corresponding eigenfunctions are foliated Schwartz symmetric. As applications we obtain the existence of nontrivial solutions for nonlinear Schrödinger equations with nonresonant nonlinearity.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Alex H. Ardila〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The aim of this paper is to provide a proof of the existence and (conditional) orbital stability for a two-parameter family of solitary-wave solutions to a coupled system of logarithmic nonlinear Schrödinger–Korteweg–de Vries equations. We also obtain the existence of global solutions for initial data in the energy space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉ℂ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉×〈/mo〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and in an appropriate Orlicz space.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Marcos Tadeu Oliveira Pimenta, Raffaella Servadei〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we consider the following nonlocal fractional variational inequality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉u〈/mi〉〈mo〉⩽〈/mo〉〈mi〉ψ〈/mi〉〈mspace width="0.16667em"〉〈/mspace〉〈mspace width="0.16667em"〉〈/mspace〉〈mtext〉a.e. in 〈/mtext〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈msub〉〈mrow〉〈mrow〉〈mo〉〈〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈mi〉λ〈/mi〉〈msub〉〈mrow〉〈mrow〉〈mo〉〈〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mo〉〉〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉⩾〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi〉Ω〈/mi〉〈/mrow〉〈/msub〉〈mi〉f〈/mi〉〈mfenced open="(" close=")"〉〈mrow〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/mfenced〉〈mrow〉〈mo〉(〈/mo〉〈mi〉v〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉−〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉d〈/mi〉〈mi〉x〈/mi〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mphantom〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈mi〉a〈/mi〉〈/mphantom〉〈mtext〉for any〈/mtext〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉v〈/mi〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mspace width="0.16667em"〉〈/mspace〉〈mi〉v〈/mi〉〈mo〉⩽〈/mo〉〈mi〉ψ〈/mi〉〈mspace width="0.16667em"〉〈/mspace〉〈mtext〉a.e. in 〈/mtext〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉Ω〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 is a smooth bounded open set with continuous boundary 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉N〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉2〈/mn〉〈mi〉s〈/mi〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mi〉λ〈/mi〉〈/math〉 is a real parameter, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mi〉f〈/mi〉〈/math〉 is function with subcritical growth, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si11.svg"〉〈mrow〉〈mi〉β〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si12.svg"〉〈mi〉ψ〈/mi〉〈/math〉 is the obstacle function. As it is well-known, the dependence of the nonlinearity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si10.svg"〉〈mi〉f〈/mi〉〈/math〉 on the term 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si14.svg"〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msup〉〈mi〉u〈/mi〉〈/mrow〉〈/math〉 makes non-variational the nature of this problem. Using an iterative technique and a penalization method, we get the existence of a nontrivial nonnegative solution for the problem under consideration, performing the Mountain Pass Theorem. This result can be seen as the extension of known existence theorem for variational inequalities driven by the Laplace operator (or more general uniformly elliptic operators) to the nonlocal fractional setting.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Anouar Bahrouni, Sabri Bahrouni, Mingqi Xiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we deal with a nonlinear problem driven by the fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mi〉M〈/mi〉〈/math〉-Laplacian and with a nonlinear nonhomogeneous reaction term. First, we give some further properties for the new fractional Orlicz–Sobolev space. Then, using the theory of nonlinear operators of monotone-type, we show the existence of a nonnegative solution. Our problem is nonvariational in nature.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): L.M. Lerman, P.E. Naryshkin, A.I. Nazarov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study entire bounded solutions to the equation 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. It is also applicable for more general equations in any dimension.〈/p〉〈/div〉

Description: 〈p〉Publication date: January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 190〈/p〉 〈p〉Author(s): Tesfa Biset, Benyam Mebrate, Ahmed Mohammed〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study boundary-value problems of Finsler infinity-Laplacian equations with nonhomogeneous terms that may exhibit singularity when solutions vanish on the boundary.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Jian Zhang, Wen Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study the following Schrödinger systems with Hardy potential 〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉u〈/mi〉〈mo〉+〈/mo〉〈mi〉u〈/mi〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mi〉v〈/mi〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉z〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉v〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈mi〉Δ〈/mi〉〈mi〉v〈/mi〉〈mo〉+〈/mo〉〈mi〉v〈/mi〉〈mo〉−〈/mo〉〈mfrac〉〈mrow〉〈mi〉μ〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉x〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mi〉u〈/mi〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉z〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉N〈/mi〉〈/mrow〉〈/msup〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉〈/span〉where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉z〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉μ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/math〉 is a positive parameter. This problem is related to coupled nonlinear Schrödinger equations for Bose–Einstein condensate. Under some suitable conditions on the parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉μ〈/mi〉〈/math〉 and nonlinearity 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mi〉f〈/mi〉〈/math〉, we first prove the existence, exponential decay and convergence of ground state solutions via variational methods. Moreover, we prove the monotonicity and convergence property of the energy of ground state solutions. Finally, we also give the asymptotic behavior of ground state solutions as parameter 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mi〉μ〈/mi〉〈/math〉 tends to 0.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 18 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis〈/p〉 〈p〉Author(s): G. Di Fazio, M.S. Fanciullo, P. Zamboni〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study a new kind of degenerate operators related to some weighted sum operators. Although the operators are severely degenerate we show the smoothness of the weak solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): T. Piasecki, Y. Shibata, E. Zatorska〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the initial–boundary value problem for the system of equations describing the flow of compressible isothermal mixture of arbitrary large number of components. The system consists of the compressible Navier–Stokes equations and a subsystem of diffusion equations for the species. The subsystems are coupled by the form of the pressure and the strong cross-diffusion effects in the diffusion fluxes of the species. Assuming the existence of solutions to the symmetrized and linearized equations, proven in Piasecki, Shibata and Zatorska (2019), we derive the estimates for the nonlinear equations and prove the local-in-time existence and maximal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/math〉 regularity of solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 189〈/p〉 〈p〉Author(s): Juan Pablo Borthagaray, Wenbo Li, Ricardo H. Nochetto〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order  〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 on a bounded domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mi〉Ω〈/mi〉〈/math〉. Such a Plateau problem of order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mi〉s〈/mi〉〈/math〉 can be reinterpreted as a Dirichlet problem for a nonlocal, nonlinear, degenerate operator of order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉. We prove that our numerical scheme converges in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mi〉r〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 for all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mi〉r〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉s〈/mi〉〈/mrow〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si7.svg"〉〈mrow〉〈msubsup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mi〉s〈/mi〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 is closely related to the natural energy space. Moreover, we introduce a geometric notion of error that, for any pair of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 functions, in the limit 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉s〈/mi〉〈mo〉→〈/mo〉〈mn〉1〈/mn〉〈mo〉∕〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/math〉 recovers a weighted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si1007.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-discrepancy between the normal vectors to their graphs. We derive error bounds with respect to this novel geometric quantity as well. In spite of performing approximations with continuous, piecewise linear, Lagrangian finite elements, the so-called 〈em〉stickiness〈/em〉 phenomenon becomes apparent in the numerical experiments we present.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Hongya Gao, Francesco Leonetti, Wei Ren〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper deals with boundary value problems of the form 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si1.svg"〉〈mfenced open="{" close=""〉〈mrow〉〈mtable align="axis" equalrows="false" columnlines="none none none" equalcolumns="false"〉〈mtr〉〈mtd columnalign="left"〉〈mo〉−〈/mo〉〈munderover〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈mi〉D〈/mi〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mi〉f〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉,〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉Ω〈/mi〉〈mo〉,〈/mo〉〈/mtd〉〈/mtr〉〈mtr〉〈mtd columnalign="left"〉〈mi〉u〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉=〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈/mtd〉〈mtd columnalign="left"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉∂〈/mi〉〈mi〉Ω〈/mi〉〈mo〉.〈/mo〉〈/mtd〉〈/mtr〉〈/mtable〉〈/mrow〉〈/mfenced〉〈/math〉 Assume that there exist 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mi〉ν〈/mi〉〈mo〉,〈/mo〉〈mi〉θ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈/math〉 such that for almost all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈mi〉x〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi〉Ω〈/mi〉〈/mrow〉〈/math〉 and all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mi〉s〈/mi〉〈mo〉,〈/mo〉〈mi〉z〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉×〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si7.svg"〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈mo〉,〈/mo〉〈mi〉z〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉|〈/mo〉〈/mrow〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mfenced open="(" close=")"〉〈mrow〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mi〉z〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈/mfenced〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mspace width="1em"〉〈/mspace〉〈mi〉i〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mo〉…〈/mo〉〈mo〉,〈/mo〉〈mi〉n〈/mi〉〈mo〉,〈/mo〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" altimg="si8.svg"〉〈mrow〉〈mi〉ν〈/mi〉〈munderover〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/munderover〉〈mfrac〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mi〉z〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mrow〉〈mo〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉+〈/mo〉〈mrow〉〈mo〉|〈/mo〉〈mi〉s〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉θ〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/mfrac〉〈mo linebreak="goodbreak" linebreakstyle="after"〉≤〈/mo〉〈munderover〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉=〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/munderover〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mrow〉〈mo〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉s〈/mi〉〈mo〉,〈/mo〉〈mi〉z〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈msub〉〈mrow〉〈mi〉z〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉.〈/mo〉〈/mrow〉〈/math〉 We let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si9.svg"〉〈mrow〉〈mi〉f〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉 and we derive regularity results for weak solutions.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Ky Ho, Yun-Ho Kim〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We obtain fundamental imbeddings for fractional Sobolev spaces with variable exponents, which are a generalization of the well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si4.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉⋅〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉-Laplacian.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Cristian E. Gutiérrez, Qingbo Huang, Henok Mawi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show existence of interfaces between two anisotropic materials so that light is refracted in accordance with a given pattern of energy. To do this we formulate a vector Snell law for anisotropic media when the wave fronts are given by norms for which the corresponding unit spheres are strictly convex.〈/p〉〈/div〉

Description: 〈p〉Publication date: October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 187〈/p〉 〈p〉Author(s): Andreas Kreuml, Olaf Mordhorst〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Dimitri Mugnai, Edoardo Proietti Lippi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We develop some properties of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mi〉p〈/mi〉〈/math〉-Neumann derivative for the fractional 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mi〉p〈/mi〉〈/math〉-Laplacian in bounded domains with general 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/math〉. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti–Rabinowitz condition.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): Dan Su, Qiaohua Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on harmonic 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si8.svg"〉〈mrow〉〈mi〉A〈/mi〉〈mi〉N〈/mi〉〈/mrow〉〈/math〉 groups 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mi mathvariant="script"〉S〈/mi〉〈/math〉 for functions whose gradient is in the Lorentz space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si3.svg"〉〈mrow〉〈mi〉L〈/mi〉〈mrow〉〈mo〉(〈/mo〉〈mi〉n〈/mi〉〈mo〉,〈/mo〉〈mi〉q〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈/mrow〉〈/math〉. Our results show that even if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si2.svg"〉〈mi mathvariant="script"〉S〈/mi〉〈/math〉 is not with strictly negative sectional curvature, one can also replace the norm 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si5.svg"〉〈mrow〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉S〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mo〉∇〈/mo〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉S〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉V〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉S〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉V〈/mi〉〈/mrow〉〈/math〉 by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" altimg="si6.svg"〉〈mrow〉〈msub〉〈mrow〉〈mo〉∫〈/mo〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉S〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mrow〉〈mo〉|〈/mo〉〈msub〉〈mrow〉〈mo〉∇〈/mo〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉S〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo〉|〈/mo〉〈/mrow〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mi〉d〈/mi〉〈mi〉V〈/mi〉〈/mrow〉〈/math〉 in the Trudinger–Moser type inequalities.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): M. Bonafini, M. Novaga, G. Orlandi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions.〈/p〉〈/div〉

Description: 〈p〉Publication date: November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 188〈/p〉 〈p〉Author(s): István Faragó, Dušan Repovš〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with the generalized Pucci–Serrin maximum principle.〈/p〉〈/div〉