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Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz–Sobolev spaces without Ambrosetti–Rabinowitz condition

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Abstract

The aim of this work is to establish the existence and multiplicity of solutions for the following class of quasilinear problems

$$\begin{aligned} - \text{ div }\big (\varepsilon ^{2}\phi (\varepsilon |\nabla u|)\nabla u\big ) + V(x)\phi (\vert u\vert )u = f(u)\quad \text{ in } \quad {\mathbb {R}}^{N}, \end{aligned}$$

where \(\varepsilon\) is a positive parameter, \(N\ge 2\), V, f are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\)-function.

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Authors and Affiliations

  1. Laboratoire “Théorie du point fixe et Applications”, École Normale Supérieure, BP 92, BP 92, 16006, Kouba, Algiers, Algeria

    Karima Ait-Mahiout

  2. Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, Campina Grande, PB, CEP:58429-900, Brazil

    Claudianor O. Alves

Authors
  1. Karima Ait-Mahiout
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  2. Claudianor O. Alves
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Correspondence to Claudianor O. Alves.

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The authors declare that there is no conflict of interest regarding the publication of this paper.

Additional information

C. O. Alves was partially supported by CNPq/Brazil Proc. 304804/2017-7.

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Ait-Mahiout, K., Alves, C.O. Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz–Sobolev spaces without Ambrosetti–Rabinowitz condition. J Elliptic Parabol Equ 4, 389–416 (2018). https://doi.org/10.1007/s41808-018-0026-1

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  • DOI: https://doi.org/10.1007/s41808-018-0026-1

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