Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space

TL;DR

This paper investigates the existence, multiplicity, and concentration phenomena of positive solutions for a class of quasilinear problems involving the $ riangle_ ext{Φ}$-Laplacian in Orlicz-Sobolev spaces, with results depending on a small parameter $ ext{ε}$.

Contribution

It establishes new results on positive solutions for quasilinear problems in Orlicz-Sobolev spaces, including multiplicity and concentration effects, extending previous work beyond standard Sobolev settings.

Findings

Existence of positive solutions under certain conditions.

Multiple positive solutions are obtained.

Solutions concentrate as the parameter ε approaches zero.

Abstract

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