Gang Li (Author), Vicenţiu Rǎdulescu (Author), Dušan Repovš (Author), Qihu Zhang (Author)

Abstract

We consider the existence of solutions of the following ▫$p(x)$▫-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: ▫$$ \begin{cases} -\text{div} \, (|\nabla u|^{p(x)-2}\nabla u) = f(x,u) & \text{ in } \quad \Omega , \\ u=0 & \text{ on } \quad \partial \Omega . \end{cases} $$▫ 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.

Keywords

nonhomogeneous differential operator;Ambrosetti-Rabinowitz condition;Cerami compactness condition;Sobolev space with variable exponent;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
UDC: 517.956.2
COBISS: 18162521 Link will open in a new window
ISSN: 1230-3429
Views: 589
Downloads: 380
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Other data

Type (COBISS): Article
Pages: str. 55-77
Volume: ǂVol. ǂ51
Issue: ǂno. ǂ1
Chronology: March 2018
DOI: 10.12775/TMNA.2017.037
ID: 11210456