# Quasilinear elliptic problem without Ambrosetti-Rabinowitz condition involving a potential in Musielak-Sobolev spaces setting

## TL;DR

This paper investigates the existence of solutions for a quasilinear elliptic problem involving a potential in Musielak-Sobolev spaces, without relying on the Ambrosetti-Rabinowitz growth condition, using variational methods.

## Contribution

It introduces a novel approach to establish solutions for elliptic problems without the classical Ambrosetti-Rabinowitz condition in Musielak-Sobolev spaces.

## Key findings

- Proves existence of weak solutions under new growth conditions.
- Extends variational methods to non-AR nonlinearities in Musielak-Sobolev spaces.
- Addresses challenges posed by variable exponent and potential functions.

## Abstract

In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(\phi(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega, \end{cases}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$), $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(\Omega)$, and $f(x,u)$ is a Carath\'eodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for $(P)$ in the framework of Musielak-Sobolev spaces. The main difficulty here is that the nonlinearity $f(x,u)$ considered does not satisfy the well-known Ambrosetti-Rabinowitz condition.