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 nonhomogeneous differential operator, Ambrosetti-Rabinowitz condition, Cerami compactness condition, Sobolev space with variable exponent Article (dk_c) 1.01 - Original Scientific Article PEF - Faculty of Education 2018 str. 55-77 Vol. 51, no. 1 517.956.2 1230-3429 10.12775/TMNA.2017.037 18162521 337 279 (0 votes) Voting is allowed only to logged in users. AddThis uses cookies that require your consent. Edit consent...

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Title: Topological Methods in Nonlinear Analysis Topol. Methods Nonlinear Anal. Juliusz Schauder Center 1230-3429 14203653