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Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition
Li, Gang (Author), Rǎdulescu, Vicenţiu (Author), Repovš, Dušan (Author), Zhang, Qihu (Author)

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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.

Language:English
Keywords:nonhomogeneous differential operator, Ambrosetti-Rabinowitz condition, Cerami compactness condition, Sobolev space with variable exponent
Work type:Article (dk_c)
Tipology:1.01 - Original Scientific Article
Organization:PEF - Faculty of Education
Year:2018
Number of pages:str. 55-77
Numbering:Vol. 51, no. 1
UDC:517.956.2
ISSN on article:1230-3429
DOI:10.12775/TMNA.2017.037 Link is opened in a new window
COBISS.SI-ID:18162521 Link is opened in a new window
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Downloads:279
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Record is a part of a journal

Title:Topological Methods in Nonlinear Analysis
Shortened title:Topol. Methods Nonlinear Anal.
Publisher:Juliusz Schauder Center
ISSN:1230-3429
COBISS.SI-ID:14203653 This link opens in a new window

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