Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

TL;DR

This paper establishes the existence and multiplicity of solutions for a variable exponent p(x)-Laplacian Dirichlet problem without relying on the classical Ambrosetti-Rabinowitz growth condition, using new growth conditions and critical point theory.

Contribution

It introduces a new growth condition that allows proving solution existence without the Ambrosetti-Rabinowitz condition, extending previous results to weaker hypotheses.

Findings

Existence of solutions under weaker growth conditions

Development of a new growth condition for Cerami compactness

Extension of previous work by Zhang and Zhao

Abstract

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: $-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }\Omega,$ and $u=0,\text{ on }\partial \Omega.$ We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. Our results extend previous work by Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comp. Math. Appl. 69 (2015), 1-12, and we establish the existence of solutions under weaker hypotheses on the nonlinear term.