## Topological Methods in Nonlinear Analysis

### Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

#### Abstract

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: \begin{equation*} \begin{cases} -{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) &\text{ in }\Omega , \\ u=0 &\text{ on }\partial \Omega . \end{cases} \end{equation*} 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. The present paper extend previous results of Q. Zhang and C. Zhao (Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Computers and Mathematics with Applications, 2015) and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 1 (2018), 55-77.

Dates
First available in Project Euclid: 18 January 2018

https://projecteuclid.org/euclid.tmna/1516244429

Digital Object Identifier
doi:10.12775/TMNA.2017.037

Mathematical Reviews number (MathSciNet)
MR3784736

Zentralblatt MATH identifier
06887972

#### Citation

Li, Gang; Rădulescu, Vicenţiu D.; Repovš, Dušan D.; Zhang, Qihu. Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition. Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 55--77. doi:10.12775/TMNA.2017.037. https://projecteuclid.org/euclid.tmna/1516244429

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