# Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

## TL;DR

This paper proves the existence of multiple solutions for a complex quasilinear elliptic problem involving the -Laplacian operator, using advanced variational methods within an Orlicz-Sobolev setting.

## Contribution

It introduces new multiplicity results for nonhomogeneous -Laplacian problems with critical growth, employing the Nehari method and concentration compactness in Orlicz spaces.

## Key findings

- Multiple solutions including a ground state established
- Application of Nehari method in Orlicz-Sobolev framework
- Overcoming challenges due to loss of homogeneity

## Abstract

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by $\Phi$-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.