Opuscula Math. 37, no. 6 (2017), 779-794

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.
  • Received: 2017-01-06.
  • Revised: 2017-01-19.
  • Accepted: 2017-01-28.
  • Published online: 2017-09-28.
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Cite this article as:
Khaled Ben Ali, Abdeljabbar Ghanmi, Khaled Kefi, On the Steklov problem involving the p(x)-Laplacian with indefinite weight, Opuscula Math. 37, no. 6 (2017), 779-794, http://dx.doi.org/10.7494/OpMath.2017.37.6.779

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