Existence of multi-peak solutions for a class of quasilinear problems in Orlicz-Sobolev spaces

TL;DR

This paper proves the existence of multiple localized solutions for a class of quasilinear elliptic equations in Orlicz-Sobolev spaces, using variational methods and under certain conditions on the involved functions.

Contribution

It establishes the existence of multi-peak solutions for quasilinear problems in Orlicz-Sobolev spaces, extending previous results to more general nonlinearities and variable potentials.

Findings

Multiple multi-peak solutions exist for the class of problems studied.

The solutions concentrate around certain points as the parameter varies.

The results apply to a broad class of nonlinearities and potentials.

Abstract

The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems \[ - \mbox{div}\big(\epsilon^{2}\phi(\epsilon|\nabla u|)\nabla u\big) + V(x)\phi(| u|)u = f(u)\quad \mbox{in} \quad \mathbb{R}^{N}, \] where $\epsilon$ is a positive parameter, $ N\geq 2$, $V, f$ are continuous functions satisfying some technical conditions and $\phi$ is a $C^{1}$-function.