ALBERT

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Richard J. Smith〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉〈em〉-locally relatively compact〈/em〉 sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications.〈/p〉〈/div〉

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: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Dekui Peng, Wei He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show in this paper that for a non-compact LCA group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈em〉τ〈/em〉 has exactly 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈/math〉 predecessors in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. This answers a problem posed in the literature affirmatively. Denote by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"〉〈mi mathvariant="fraktur"〉N〈/mi〉〈/math〉 the class of all LCA groups 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is precompact. It is shown that for an LCA group 〈em〉G〈/em〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si207.svg"〉〈mi〉G〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="fraktur"〉N〈/mi〉〈/math〉 if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si212.svg"〉〈mi〉G〈/mi〉〈mo〉≅〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈mi〉H〈/mi〉〈/math〉, where 〈em〉n〈/em〉 is a non-negative integer and 〈em〉H〈/em〉 is an LCA group with an open compact subgroup 〈em〉N〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si213.svg"〉〈mi〉H〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉N〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉. As an application of this result, we extend a well known result on discrete abelian groups to the case of LCA groups. We show that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a non-compact LCA group and 〈em〉σ〈/em〉 is a predecessor of 〈em〉τ〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, then the connected component of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 coincides with the connected component of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si274.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉σ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. It is also shown that for a non-compact LCA group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, if 〈em〉σ〈/em〉 is a predecessor of 〈em〉τ〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, then the equality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉σ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 holds. This partially answer a question posed in the literature.〈/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: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Violeta Vasilevska〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper provides further investigation of the concept of shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉-fibrators (previously introduced by the author). The main results identify shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉-fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si209.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 with coperfectly Hopfian group (a new class of Hopfian groups that are introduced here) is a shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉o-fibrator if it is a codimension-2 fibrator (Theorem 5.4). The main result (Theorem 6.2) states that the direct product of two closed orientable manifolds (of different dimension) homotopically determined by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si209.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and with coperfectly Hopfian fundamental groups (one normally incommensurable with the other one) is a shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉o-fibrator, if it is a Hopfian manifold and a codimension-2 fibrator.〈/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: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Han Lou, Mingxing Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉M〈/em〉 be a simple 3-manifold, and 〈em〉F〈/em〉 be a component of ∂〈em〉M〈/em〉 of genus at least 2. Let 〈em〉α〈/em〉 and 〈em〉β〈/em〉 be separating slopes on 〈em〉F〈/em〉. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 (resp. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉β〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉) be the manifold obtained by adding a 2-handle along 〈em〉α〈/em〉 (resp. 〈em〉β〈/em〉). If 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉β〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are ∂-reducible, then the minimal geometric intersection number of 〈em〉α〈/em〉 and 〈em〉β〈/em〉 is at most 8.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Keita Nakagane〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show, using Mellit's recent results, that Kálmán's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Jerzy Ka̧kol〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Following Christensen [11] a metrizable space 〈em〉X〈/em〉 is 〈em〉σ〈/em〉-compact if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is analytic, i.e. it is a continuous image of the Polish space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉. By Michael [26] the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous image of a separable metric space if and only if 〈em〉X〈/em〉 is cosmic, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉ℵ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉-space if and only if 〈em〉X〈/em〉 is countable. We show here that, in parallel manner, Christensen-Calbrix and Michael results may be characterized as follows: For a metrizable space 〈em〉X〈/em〉 the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is analytic if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous linear image of a separable metrizable locally convex space, and a Tychonoff space 〈em〉X〈/em〉 is a submetrizable hemicompact 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/msub〉〈/math〉-space if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous linear image of a metrizable and complete separable locally convex space. Applications are provided.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Dekui Peng, Wei He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A Hausdorff topological group 〈em〉G〈/em〉 is called 〈em〉lower continuous〈/em〉 if the topology of 〈em〉G〈/em〉 has no predecessor in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. The class of lower continuous topological groups contains all closed subgroups of products of minimal abelian groups, so strictly extend the class of minimal groups. Our main concern in this paper is the study of properties of lower continuous topological groups. Similar with the case for minimal groups, we provide a lower continuity criterion: a dense subgroup 〈em〉H〈/em〉 of a Hausdorff topological abelian group 〈em〉G〈/em〉 is lower continuous if and only if 〈em〉G〈/em〉 is lower continuous and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"〉〈mi〉S〈/mi〉〈mi〉o〈/mi〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉H〈/mi〉〈/math〉. It is shown that every totally lower continuous abelian group is precompact. It is also shown that for a compact abelian groups 〈em〉G〈/em〉, 〈em〉G〈/em〉 is hereditarily lower continuous if and only if 〈em〉G〈/em〉 is torsion-free.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Javier Camargo, Sergio Macías, Marco Ruiz〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study three different topics related to the Jones' set function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉. The first topic is idempotency; we study differences between idempotency on continua and idempotency on closed sets. The second aspect that we present are some properties about the set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. Particularly, we show that it is not possible to find a continuum 〈em〉X〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is compact and countable; and we give a continuum 〈em〉X〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is countable. Finally, the third topic is the behavior of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉 on products. One of our main results is that the compactness of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi〉Y〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 implies the local connectedness of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi〉Y〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Wei-Feng Xuan, Yan-Kui Song〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The notion of a 〈em〉κ〈/em〉-splitting diagonal was introduced and studied by Tkachuk. In this paper, we prove that there exists a locally countable, locally compact space with an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉-splitting diagonal but no 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal. Using the Erdös-Radó's theorem, we also prove that every DCCC homogeneous space 〈em〉X〈/em〉 with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"〉〈mi〉π〈/mi〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉ω〈/mi〉〈/math〉 has cardinality at most 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"〉〈mi mathvariant="fraktur"〉c〈/mi〉〈/math〉. Some new questions are also posed.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): S. Cobzaş〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We prove versions of Ekeland, Takahashi and Caristi principles in sequentially right 〈em〉K〈/em〉-complete quasi-pseudometric spaces (meaning asymmetric pseudometric spaces), the equivalence between these principles, as well as their equivalence to the completeness of the underlying quasi-pseudometric space.〈/p〉 〈p〉The key tools are Picard sequences for some special set-valued mappings corresponding to a function 〈em〉φ〈/em〉 on a quasi-pseudometric space, allowing a unitary treatment of all these principles.〈/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: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): W. Xi, D. Dikranjan, M. Shlossberg, D. Toller〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We study locally compact groups having all dense subgroups (locally) minimal. We call such groups 〈em〉densely (locally) minimal〈/em〉. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 of 〈em〉p〈/em〉-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups.〈/p〉 〈p〉We prove that in case that a topological abelian group 〈em〉G〈/em〉 is either compact or connected locally compact, then 〈em〉G〈/em〉 is densely locally minimal if and only if 〈em〉G〈/em〉 either is a Lie group or has an open subgroup isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 for some prime 〈em〉p〈/em〉. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Oorna Mitra, Parameswaran Sankaran〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that certain groups of diffeomorphisms and PL-homeomorphisms embed in the group of all quasi-isometry classes of the Euclidean spaces.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Dániel T. Soukup, Paul J. Szeptycki〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A topological space 〈em〉X〈/em〉 is strongly 〈em〉D〈/em〉 if for any neighbourhood assignment 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, there is a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉D〈/mi〉〈mo〉⊆〈/mo〉〈mi〉X〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉D〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 covers 〈em〉X〈/em〉 and 〈em〉D〈/em〉 is locally finite in the topology generated by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉. We prove that ⋄ implies that there is an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si308.svg"〉〈mrow〉〈mi mathvariant="normal"〉HF〈/mi〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉w〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/math〉 space in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈/math〉 (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly 〈em〉D〈/em〉. We also show that any HFC space 〈em〉X〈/em〉 is dually discrete and if additionally countable sets have Menger closure then 〈em〉X〈/em〉 is a 〈em〉D〈/em〉-space.〈/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: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Bingzhe Hou, Geng Tian〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we introduce some new equivalence relations for topological dynamical systems named strong topological shift equivalence and topological shift equivalence, which are similar to the strong shift equivalence and shift equivalence for subshifts of finite type. We study the relations between the new equivalences and other equivalences such as topological conjugacy, mutually topological semi-conjugacy and canonical homeomorphism extensions being topologically conjugate. Some properties and examples are shown. In particular, mean topological dimension is an invariant for topological shift equivalence but not for mutually topologically semi-conjugate equivalence. In this topic, linear operators are also considered.〈/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: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Sajjad Mohammadi, Mohammad A. Asadi-Golmankhaneh〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be the principal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉S〈/mi〉〈mi〉U〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn〉4〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-bundle over 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉8〈/mn〉〈/mrow〉〈/msup〉〈/math〉 with Chern class 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉6〈/mn〉〈mi〉k〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be the gauge group of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 classified by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉k〈/mi〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉 a generator of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉8〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉B〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mi〉U〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn〉4〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≅〈/mo〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/math〉. In this article we partially classify the homotopy types of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 by showing that if there is a homotopy equivalence 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo〉≃〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈/math〉 then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉420〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉420〈/mn〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉3360〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is equal to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉3360〈/mn〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo〉≃〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈/math〉.〈/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: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): A. Bartoš, J. Bobok, J. van Mill, P. Pyrih, B. Vejnar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce the notion of compactifiable classes – these are classes of metrizable compact spaces that can be up to homeomorphic copies “disjointly combined” into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions – we give several characterizations, consider preservation under various constructions, and raise several questions.〈/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: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Carolina Medina, Gelasio Salazar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉L〈/em〉 be a fixed link. Given a link diagram 〈em〉D〈/em〉, is there a sequence of crossing exchanges and smoothings on 〈em〉D〈/em〉 that yields a diagram of 〈em〉L〈/em〉? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if 〈em〉L〈/em〉 is a prime link with crossing number at most 5, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus links 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉m〈/mi〉〈/mrow〉〈/msub〉〈/math〉 and all twist knots.〈/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: Available online 27 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Akira Koyama, Vesko Valov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide some properties and characterizations of homologically 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉U〈/mi〉〈msup〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉-maps and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉l〈/mi〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉-spaces. We show that there is a parallel between recently introduced by Cauty [3] algebraic 〈em〉ANR〈/em〉's and homologically 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉l〈/mi〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉-metric spaces, and this parallel is similar to the parallel between ordinary 〈em〉ANR〈/em〉's and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉-metric spaces. We also show that there is a similarity between the properties of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉-spaces and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉l〈/mi〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉-spaces. Some open questions are raised.〈/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: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Henry Adams, Joshua Mirth〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given a sample 〈em〉Y〈/em〉 from an unknown manifold 〈em〉X〈/em〉 embedded in Euclidean space, it is possible to recover the homology groups of 〈em〉X〈/em〉 by building a Vietoris–Rips or Čech simplicial complex on top of the vertex set 〈em〉Y〈/em〉. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when 〈em〉Y〈/em〉 is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the 〈em〉Vietoris–Rips〈/em〉 and 〈em〉Čech thickenings〈/em〉, which are equipped with the 1-Wasserstein metric in place of the simplicial complex topology. We show that for Euclidean subsets 〈em〉X〈/em〉 with positive reach, the thickenings satisfy metric analogues of Hausmann's theorem and the nerve lemma (the metric Vietoris–Rips and Čech thickenings of 〈em〉X〈/em〉 are homotopy equivalent to 〈em〉X〈/em〉 for scale parameters less than the reach). To our knowledge this is the first version of Hausmann's theorem for Vietoris–Rips constructions on entire Euclidean submanifolds (as opposed to Riemannian manifolds), and our result also extends to non-manifold shapes (as not all sets of positive reach are manifolds). In contrast to Hausmann's original proof, our homotopy equivalence is a deformation retraction, is realized by canonical maps in both directions, and furthermore can be proven to be a homotopy equivalence via simple linear homotopies from the map compositions to the corresponding identity maps.〈/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: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano, Christopher Mouron〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we show that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉h〈/mi〉〈mo〉:〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi〉X〈/mi〉〈/math〉 is a mixing homeomorphism on a 〈em〉G〈/em〉-like continuum, then 〈em〉X〈/em〉 must be indecomposable and if 〈em〉X〈/em〉 is finitely cyclic, then 〈em〉X〈/em〉 must be 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉-indecomposable for some natural number 〈em〉n〈/em〉. Furthermore, we give an example of a mixing homeomorphism on a hereditary decomposable tree-like continuum.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Fortunata Aurora Basile, Maddalena Bonanzinga, Nathan Carlson, Jack Porter〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we extend the theory of H-closed extensions of Hausdorff spaces to a class of non-Hausdorff spaces, defined in [2], called 〈em〉n〈/em〉-Hausdorff spaces. The notion of H-closed is generalized to an 〈em〉n〈/em〉-H-closed space. Known construction for Hausdorff spaces 〈em〉X〈/em〉, such as the Katětov H-closed extension 〈em〉κX〈/em〉, are generalized to a maximal 〈em〉n〈/em〉-H-closed extension denoted by 〈em〉n〈/em〉-〈em〉κX〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Akram Alishahi, Nathan Dowlin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that the page at which the Lee spectral sequence collapses gives a bound on the unknotting number, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉K〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. In particular, for knots with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉K〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈/math〉, we show that the Lee spectral sequence must collapse at the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉E〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉 page. An immediate corollary is that the Knight Move Conjecture is true when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉K〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Urtzi Buijs, José M. Moreno-Fernández〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide two criteria for establishing the non-formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogues of the Massey products of a differential graded associative algebra. We also show that formality of a differential graded Lie algebra is not equivalent to the collapse of its associated Quillen spectral sequence. Finally, we use 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msub〉〈/math〉 algebras and Quillen's formulation of rational homotopy theory to recover and improve a classical theorem for detecting higher Whitehead products in Sullivan minimal models, and give some applications.〈/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: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Michael Albanese, Aleksandar Milivojević〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We record an answer to the question “In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?”. In the process of doing so, we are naturally led to ask “For which values of 〈em〉ℓ〈/em〉 is the connected sum of 〈em〉ℓ〈/em〉 closed almost complex manifolds necessarily an almost complex manifold?”. We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-connected 2〈em〉n〈/em〉-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mtext〉spin〈/mtext〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈/msup〉〈/math〉 Dirac operator.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Henry Jose Gullo Mercado, Leandro Fiorini Aurichi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be a topological space and let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉 be the family of all subsets of 〈em〉X〈/em〉 that satisfy a given topological property 〈em〉P〈/em〉 (invariant under homeomorphisms). If we add new open sets to the topology and if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi mathvariant="script"〉F〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉 is the family of all subsets of the new space which satisfy the property 〈em〉P〈/em〉, we can have 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈mo〉≠〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="script"〉F〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉. If this is always the case, we say that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is maximal with respect to the family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉. We show here some characterizations of maximal spaces with respect to the family of discrete subsets.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Cerene Rathilal, Dharmanand Baboolal, Paranjothi Pillay〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given a locally connected metric frame 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉L〈/mi〉〈mo〉,〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 a new compatible metric diameter on 〈em〉L〈/em〉 is shown to exist, which in the spatial case corresponds to a metric due to Kelley [8] having the property that all spherical neighbourhoods of a point are connected and have property S. Our main result is that if a locally connected metric frame has property S then its top element can be written as finite join of arbitrarily small connected elements each having property S.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Maria Trnková〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we address some problems concerning an approximate Dirichlet domain. We show that under some assumptions an approximate Dirichlet domain can work equally well as an exact Dirichlet domain. In particular, we consider a problem of tiling a hyperbolic ball with copies of the Dirichlet domain. This problem arises in the construction of the length spectrum algorithm which is implemented by the computer program SnapPea. Our result explains the empirical fact that the program works surprisingly well despite it does not use exact data. Also we demonstrate a rigorous verification whether two words of the fundamental group of a hyperbolic 3-manifold are the same or not.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Zava Nicolò〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider two categories: the category 〈strong〉Coarse〈/strong〉 of coarse spaces and bornologous maps and its quotient category 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="bold"〉Coarse〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉/〈/mo〉〈/mrow〉〈mrow〉〈mo〉∼〈/mo〉〈/mrow〉〈/msub〉〈/math〉, where ∼ is the closeness relation. This paper tackles the problem of their wellpoweredness and cowellpoweredness. In particular, we show that all the epireflective subcategories of 〈strong〉Coarse〈/strong〉 are cowellpowered, using a complete characterisation of closure operators of 〈strong〉Coarse〈/strong〉, while 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="bold"〉Coarse〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉/〈/mo〉〈/mrow〉〈mrow〉〈mo〉∼〈/mo〉〈/mrow〉〈/msub〉〈/math〉 is both wellpowered and cowellpowered. Moreover, we prove that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="bold"〉Coarse〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉/〈/mo〉〈/mrow〉〈mrow〉〈mo〉∼〈/mo〉〈/mrow〉〈/msub〉〈/math〉 has neither equalizers nor pullbacks of subobjects, although it has arbitrary products.〈/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: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Andrzej Kucharski, Sławomir Turek〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce a new class of 〈em〉ϰ〈/em〉-metrizable spaces, namely countably 〈em〉ϰ〈/em〉-metrizable spaces. We show that the class of all 〈em〉ϰ〈/em〉-metrizable spaces is a proper subclass of countably 〈em〉ϰ〈/em〉-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with 〈em〉ϰ〈/em〉-metrizable spaces. We prove a generalization of Chigogidze's result that the Čech-Stone compactification of a pseudocompact countably 〈em〉ϰ〈/em〉-metrizable space is 〈em〉ϰ〈/em〉-metrizable.〈/p〉〈/div〉

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: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Angelo Bella, Alan Dow〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Few observations on a paper of Arhangel'skiĭ and Buzyakova led us to consider Rančin's problem. The main result here is the construction under ⋄ of a compact c-sequential space that is not sequential.〈/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: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: 1 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 268〈/p〉 〈p〉Author(s): Angelo Bella〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The inequality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉 has been proved to be true for both the class of Lindelöf spaces (Arhangel'skii, 1969 [1]) and that of 〈em〉H〈/em〉-closed spaces (Dow and Porter, 1982 [6]), by different arguments. We present a common weakening of the Lindelöf and 〈em〉H〈/em〉-closed properties which allows us to give a unified proof of these two theorems.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 268〈/p〉 〈p〉Author(s): Jaume Llibre, Víctor F. Sirvent〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the present article we give sufficient conditions for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉 self-maps on some connected compact manifolds in order to have positive entropy. The conditions are given in terms of the Lefschetz numbers of the iterates of the map and/or its Lefschetz zeta function. We consider the cases where the manifold is a compact orientable and non-orientable surface, the 〈em〉n〈/em〉-dimensional torus, the product of 〈em〉n〈/em〉 spheres of dimension 〈em〉ℓ〈/em〉 and the product of spheres of different dimensions.〈/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: 1 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 268〈/p〉 〈p〉Author(s): Lev Bukovský, Alexander V. Osipov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉USC〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be the topological space of real upper semicontinuous bounded functions defined on 〈em〉X〈/em〉 with the subspace topology of the product topology on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mmultiscripts〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mprescripts〉〈/mprescripts〉〈none〉〈/none〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/mmultiscripts〉〈/math〉. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Φ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ψ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈/math〉 are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉USC〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, respectively. We prove several equivalent assertions to that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉USC〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 satisfies the selection principles S〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Φ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ψ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, including a condition on the topological space 〈em〉X〈/em〉.〈/p〉 〈p〉We prove similar results for the topological space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of continuous bounded functions.〈/p〉 〈p〉Similar results hold true for the selection principles S〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msub〉〈mrow〉〈/mrow〉〈mrow〉〈mi〉f〈/mi〉〈mi〉i〈/mi〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Φ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ψ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/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: 1 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 270〈/p〉 〈p〉Author(s): Volodymyr Mykhaylyuk, Vadym Myronyk〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We analyze compactness-like properties of sets in partial metric spaces and obtain the equivalence of several forms of the compactness for partial metric spaces. Moreover, we give a negative answer to a question from [8] on the existence of a bounded complete partial metric on a complete partial metric space.〈/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: Available online 7 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Salvador Hernández, F. Javier Trigos-Arrieta〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a Hausdorff precompact abelian group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="normal"〉w〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of the topology w. As an application, an example of a MAP group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is given such that every closed, metrizable subgroup 〈em〉N〈/em〉 of 〈em〉bG〈/em〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉∩〈/mo〉〈mi〉G〈/mi〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 preserves compactness but 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 does not strongly respect compactness. Thereby, we respond to Questions 4.1 and 4.3 in [6].〈/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: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 257〈/p〉 〈p〉Author(s): Daron Anderson, Paul Bankston〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We import into continuum theory the notion of 〈em〉extreme point of a convex set〈/em〉 from the theory of topological vector spaces. We explore how extreme points relate to other established types of “edge point” of a continuum; for example we prove that extreme points are always shore points, and that any extreme point is also non-block if the continuum is either decomposable or irreducible (in particular, metrizable).〈/p〉 〈p〉In addition we discuss some continuum-theoretic analogues of the celebrated Krein-Milman theorem.〈/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: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Mohammad Javaheri〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that if a homeomorphism of a separable locally compact metric space has a unique fixed point that is attracting or repelling, then its corresponding composition operator is cyclic. On the real line, we show that a composition operator is cyclic if and only if its symbol has at most one fixed point. Other results on the real line and circle are also discussed.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 257〈/p〉 〈p〉Author(s): Taketo Shirane〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the 〈em〉splitting graph〈/em〉. By using the splitting graph, we classify the embedded topology of plane curves consisting of one smooth curve and non-concurrent three lines, called 〈em〉Artal arrangements〈/em〉.〈/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: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Xin Liu, Chuan Liu, Shou Lin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Based on the notions of T. Banakh's strict Pytkeev networks and A.V. Arhangel'skiı̌'s sensor families, strict Pytkeev networks with sensors are introduced in this paper. A family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉P〈/mi〉〈/math〉 of subsets of a topological space 〈em〉X〈/em〉 is called a 〈em〉strict Pytkeev network with sensors〈/em〉 (abbr. an 〈em〉sp-network〈/em〉) if, for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉U〈/mi〉〈mo〉∩〈/mo〉〈mover accent="true"〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈/math〉 with 〈em〉U〈/em〉 open and 〈em〉A〈/em〉 subset in 〈em〉X〈/em〉, there is a set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉P〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉P〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉P〈/mi〉〈mo〉⊂〈/mo〉〈mi〉U〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.gif" overflow="scroll"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mover accent="true"〉〈mrow〉〈mi〉P〈/mi〉〈mo〉∩〈/mo〉〈mi〉A〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈/math〉. In present paper, we discuss certain relationship and operations among spaces defined by special Pytkeev networks, study spaces with a point-countable 〈em〉sp〈/em〉-network and spaces with a 〈em〉σ〈/em〉-closure-preserving 〈em〉sp〈/em〉-network, and detect some applications of 〈em〉sp〈/em〉-networks in topological groups.〈/p〉 〈p〉The following results are obtained: (1) Every 〈em〉sp〈/em〉-network is preserved by a continuous pseudo-open mapping. (2) Every 〈em〉k〈/em〉-space with a point-countable 〈em〉sp〈/em〉-network coincides with a continuous pseudo-open 〈em〉s〈/em〉-image of a metric space. (3) Every regular feebly compact space with a point-countable 〈em〉sp〈/em〉-network has a point-countable base. (4) A regular space has a countable 〈em〉sp〈/em〉-network if and only if it is separable and has a point-countable 〈em〉sp〈/em〉-network. (5) A topological space is stratifiable if and only if it is a regular space with a 〈em〉σ〈/em〉-closure-preserving 〈em〉sp〈/em〉-network. (6) A regular space with a 〈em〉σ〈/em〉-locally finite 〈em〉sp〈/em〉-network has a 〈em〉σ〈/em〉-discrete 〈em〉sp〈/em〉-network. (7) A topological group is metrizable if it has countable 〈em〉sp〈/em〉-character. (8) There is a non-Fréchet-Urysohn sequential topological group with a countable strict Pytkeev network, which give a negative answer to a question posed by A.V. Arhangel'skiı̌ [1].〈/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: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Mostafa Abedi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉 be the ring of continuous real-valued functions on a completely regular frame 〈em〉L〈/em〉. We study the class of prime ideals 〈em〉P〈/em〉 of the ring 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉 determined by the condition: 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉P〈/mi〉〈/math〉 is a real-closed ring. We give some necessary and sufficient frame-theoretic conditions for an ordered integral domain of the form 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉P〈/mi〉〈/math〉 to be a real-closed ring, when 〈em〉P〈/em〉 is a prime 〈em〉z〈/em〉-ideal (〈em〉d〈/em〉-ideal) of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉. A completely regular frame 〈em〉L〈/em〉 is called an 〈em〉SV〈/em〉-frame if for every prime ideal 〈em〉P〈/em〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉, the ordered integral domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉P〈/mi〉〈/math〉 is a real-closed ring. It is shown that every 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si235.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-quotient of an 〈em〉SV〈/em〉-frame is an 〈em〉SV〈/em〉-frame. We also show that open quotients ↓〈em〉c〈/em〉 in an 〈em〉SV〈/em〉-frame are 〈em〉SV〈/em〉-frames for all cozero elements 〈em〉c〈/em〉. Larson [22] has given a topological characterization of compact 〈em〉SV〈/em〉-spaces. By extending this characterization to frames, we show that the compactness limitation can really be relaxed, even in spaces, and so strengthen Larson's result.〈/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: Available online 20 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Chi-Keung Ng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉E〈/mi〉〈/math〉 on a set 〈em〉X〈/em〉 is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of non-empty subsets of 〈em〉X〈/em〉 and show that it induces the Hausdorff coarse structure on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉.〈/p〉 〈p〉On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 on a set 〈em〉X〈/em〉 is induced by a map 〈em〉d〈/em〉 from 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi〉X〈/mi〉〈/math〉 to a partially ordered set (with no requirement on 〈em〉d〈/em〉) if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 admits a base 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.gif" overflow="scroll"〉〈mi mathvariant="script"〉B〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.gif" overflow="scroll"〉〈mi mathvariant="script"〉B〈/mi〉〈mo〉∪〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mo〉⋂〈/mo〉〈mi mathvariant="script"〉U〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 is closed under arbitrary intersections. In this case, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 is actually defined by a pseudo uniform metric. We also show that a uniform structures 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 comes from a pseudo uniform metric that takes values in a totally ordered set if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 admits a totally ordered base.〈/p〉 〈p〉Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set.〈/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: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Héctor Barge, José M.R. Sanjurjo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study continuous parametrized families of dissipative flows, which are those flows having a global attractor. The main motivation for this study comes from the observation that, in general, global attractors are not robust, in the sense that small perturbations of the flow can destroy their globality. We give a necessary and sufficient condition for a global attractor to be continued to a global attractor. We also study, using shape theoretical methods and the Conley index, the bifurcation global to non-global.〈/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: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Javier Camargo, Mayer Palacios, Hugo Villanueva〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study when a strongly freely decomposable mapping is almost monotone. Also, we define the notion of 〈em〉i〈/em〉-unicoherent continuum and present some properties and examples of this class of continua.〈/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: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Kyeonghui Lee, Young Ho Im, Sera Kim〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce a family of polynomial invariants for flat virtual knots which extend the polynomial introduced by Kauffman and Richter, and we give several properties and examples.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Kengo Kawamura〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉An embedded/immersed surface-knot is a closed and connected surface embedded/immersed in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈/math〉, respectively. The triple point number 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of an embedded/immersed surface-knot 〈em〉F〈/em〉 is the minimum number of triple points required for a diagram of 〈em〉F〈/em〉. Satoh proved that (i) there does not exist an embedded surface-knot 〈em〉F〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/math〉, and (ii) there does not exist an embedded 2-knot 〈em〉F〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉2〈/mn〉〈mtext〉 or 〈/mtext〉〈mn〉3〈/mn〉〈/math〉. In this paper, we prove similar results for immersed surface-knots with some conditions.〈/p〉〈/div〉