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Multiplicity and concentration behavior of positive solutions for a quasilinear problem in Orlicz–Sobolev spaces without Ambrosetti–Rabinowitz condition via penalization method

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Abstract

This work is concerned with existence, multiplicity and concentration of positive solutions for the following class of quasilinear problems

$$\begin{aligned} -\Delta _{\Phi }u+V(\varepsilon x)\phi (|u|) u=f(u)\quad \text {in} ~ {\mathbb {R}}^N~(N\ge 2), \end{aligned}$$

where \(\Phi (t)=\int _0^{|t|}\phi (s)sds\) is a N-function, \(\Delta _{\Phi }\) is the \(\Phi\)-Laplacian operator, \(\varepsilon\) is a positive parameter, \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a continuous function and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a \(C^1\)-function.

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Acknowledgements

I would like to thank the anonymous referees for their careful reading and helpful suggestions which led to a substantial improvement of the original manuscript.

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

    K. Ait-Mahiout

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Ait-Mahiout, K. Multiplicity and concentration behavior of positive solutions for a quasilinear problem in Orlicz–Sobolev spaces without Ambrosetti–Rabinowitz condition via penalization method. J Elliptic Parabol Equ 6, 473–506 (2020). https://doi.org/10.1007/s41808-020-00054-0

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