Multiple solutions for a class of quasilinear problems in Orlicz–Sobolev spaces

Abstract

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$\begin{array}{c}-{\mathrm{\Delta }}_{\mathrm{\Phi }}\mathit{u}+\mathit{\varphi }\mathrm{(}|\mathit{u}|\mathrm{)}\mathit{u}=\mathit{h}\mathrm{(}\mathit{\epsilon }\mathit{x}\mathrm{)}\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}\text{in }{\mathbb{R}}^{\mathit{N}}\mathrm{,}\mathit{u}\in {\mathit{W}}^{1\mathrm{,}\mathrm{\Phi }}\mathrm{\left(}{\mathbb{R}}^{\mathit{N}}\mathrm{\right)}\mathrm{,}\end{array}$$ where $\mathrm{\Phi }\mathrm{(}\mathit{t}\mathrm{)}={\int }_0^{|\mathit{t}|}\mathit{\varphi }\mathrm{(}\mathit{s}\mathrm{)}\mathit{s}\mathit{d}\mathit{s}$ is an N-function, ${\mathrm{\Delta }}_{\mathrm{\Phi }}$ is the Φ-Laplacian operator, $\mathit{\epsilon }\mathrm{>}0$ and $\mathit{h}\mathrm{,}\mathit{f}$ are continuous functions. In the proof of our main result we have used Variational methods, Ekeland’s variational principle and some properties of the Nehari manifolds.