Multiplicity and concentration behavior of solutions for a quasilinear problem involving $N$-functions via penalization method

Abstract

In this work, we study the existence, multiplicity and concentration of positive solutions for the following class of quasilinear problem: \[ - \Delta_{\Phi}u + V(\epsilon x)\phi(\vert u\vert)u = f(u)\quad \mbox{in} \quad \mathbb{R}^{N}, \] where $\Phi(t) = \int_{0}^{\vert t\vert}\phi(s)sds$ is a N-function, $ \Delta_{\Phi}$ is the $\Phi$-Laplacian operator, $\epsilon$ is a positive parameter, $ N\geq 2$, $V : \mathbb{R}^{N} \rightarrow \mathbb{R} $ is a continuous function and $f : \mathbb{R} \rightarrow \mathbb{R} $ is a $C^{1}$-function.