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On a Class of Quasilinear Equations Involving Critical Exponential Growth and Concave Terms in \({\mathbb {R}}^N\)

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Abstract

In this work, we establish the existence and multiplicity of nonzero and non-negative solutions for a class of quasilinear elliptic equations, whose nonlinearity is allowed to enjoy the critical exponential growth with respect to a version of the Trudinger–Moser inequality, and it can also contain concave terms in \({\mathbb {R}}^N (N\ge 2)\). When \(N=2\) or \(N=3\), this equation is motivated for a physics application in fluid mechanics. In order to obtain our results, we combine variational arguments in a suitable subspace of a Orlicz–Sobolev space with a version of the Trudinger–Moser inequality and Ekeland variational principle. In a particular case, we show that the solution is a positive ground state.

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Notes

  1. A version for bounded domains can be found for instance in [15, 25].

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Authors and Affiliations

  1. Universidade Federal de Campina Grande Unidade Acadêmica de Matemática, Campina Grande, PB, 58109-970, Brazil

    Jefferson Abrantes Santos

  2. Universidade Federal da Paraíba, Departamento de Matemática, João Pessoa, PB, 58051-900, Brazil

    Uberlandio B. Severo

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  1. Jefferson Abrantes Santos
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  2. Uberlandio B. Severo
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Correspondence to Jefferson Abrantes Santos.

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Communicated by Nader Masmoudi.

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Research partially supported by CNPq-Brazil Grant Casadinho/Procad 552.464/2011-2.

Research partially supported by CNPq Grant 310747/2019-8 and Grant 2019/0014 Paraíba State Research Foundation (Fapesq), Brazil.

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Santos, J.A., Severo, U.B. On a Class of Quasilinear Equations Involving Critical Exponential Growth and Concave Terms in \({\mathbb {R}}^N\). Ann. Henri Poincaré 23, 1–24 (2022). https://doi.org/10.1007/s00023-021-01054-z

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  • DOI: https://doi.org/10.1007/s00023-021-01054-z

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