Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

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

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
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2018
• Number of pages: Str. 55-77
• Numbering: Vol. 51, no. 1
• PID: 20.500.12556/RUL-109292
• UDC: 517.956.2
• ISSN on article: 1230-3429
• DOI: 10.12775/TMNA.2017.037
• COBISS.SI-ID: 18162521
• Publication date in RUL: 29.08.2019

LI, Gang, RǍDULESCU, Vicenţiu, REPOVŠ, Dušan and ZHANG, Qihu, 2018, Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition. Topological Methods in Nonlinear Analysis [online]. 2018. Vol. 51, no. 1, p. 55–77. [Accessed 17 April 2025]. DOI 10.12775/TMNA.2017.037. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=109292

Record is a part of a journal

• Title: Topological Methods in Nonlinear Analysis
• Shortened title: Topol. Methods Nonlinear Anal.
• Publisher: Juliusz Schauder Center for Nonlinear Studies
• ISSN: 1230-3429
• COBISS.SI-ID: 14203653