Opuscula Math. 37, no. 6 (2017), 779-794
http://dx.doi.org/10.7494/OpMath.2017.37.6.779

Opuscula Mathematica

On the Steklov problem involving the p(x)-Laplacian with indefinite weight

Khaled Ben Ali
Abdeljabbar Ghanmi
Khaled Kefi

Abstract. Under suitable assumptions, we study the existence of a weak nontrivial solution for the following Steklov problem involving the $$p(x)$$-Laplacian $\begin{cases}\Delta_{p(x)}u=a(x)|u|^{p(x)-2}u \quad \text{in }\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda V(x)|u|^{q(x)-2}u \quad \text{on }\partial \Omega.\end{cases}$ Our approach is based on min-max method and Ekeland's variational principle.

Keywords: $$p(x)$$-Laplace operator, Steklov problem, variable exponent Sobolev spaces, variational methods, Ekeland's variational principle.

Mathematics Subject Classification: 35J48, 35J66.

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• Khaled Ben Ali
• Jazan Technical College, P.O. Box: 241 Jazan 45952, Saudi Arabia
• University of Tunis El Manar, Faculty of Sciences, 1060 Tunis, Tunisia
• Abdeljabbar Ghanmi
• University of Jeddah, Faculty of Science and Arts, Mathematics Department, Khulais, Saudi Arabia
• University of Tunis El Manar, Faculty of Sciences, 1060 Tunis, Tunisia
• Khaled Kefi
• Northern Border University, Community College of Rafha, Saudi Arabia
• University of Tunis El Manar, Faculty of Sciences, 1060 Tunis, Tunisia
• Communicated by Vicentiu D. Radulescu.
• Revised: 2017-01-19.
• Accepted: 2017-01-28.
• Published online: 2017-09-28.