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

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 $Φ(t) = \int_{0}^{\vert t\vert}ϕ(s)sds$ is a N-function, $Δ_Φ$ is the $Φ$-Laplacian operator, $ε$ 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.

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