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: \[ - Δ_Φu + V(εx)ϕ(\vert u\vert)u = f(u)\quad \mbox{in} \quad \mathbb{R}^{N}, \] where $\Phi (t)={\int }_0^{|t|}\varphi (s)sds$ is a N-function, ${\Delta }_{\Phi }$ is the $\Phi$-Laplacian operator, $\epsilon$ is a positive parameter, $N\ge 2$, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous function and $f:\mathbb{R}\to \mathbb{R}$ is a $C^1$-function.

arXiv admin note: text overlap with arXiv:1506.01669