Opuscula Math. 35, no. 6 (2015), 853-866

Opuscula Mathematica

Continuous spectrum of Steklov nonhomogeneous elliptic problem

Mostafa Allaoui

Abstract. By applying two versions of the mountain pass theorem and Ekeland's variational principle, we prove three different situations of the existence of solutions for the following Steklov problem: \[\begin{aligned}\Delta_{p(x)} u&=|u|^{p(x)-2}u \phantom{\lambda} \quad\text{in}\;\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}&= \lambda|u|^{q(x)-2}u \quad\text{on}\;\partial\Omega,\end{aligned}\] where \(\Omega \subset \mathbb{R}^N\) \((N\geq 2)\) is a bounded smooth domain and \(p,q: \overline{\Omega}\rightarrow(1,+\infty)\) are continuous functions.

Keywords: \(p(x)\)-Laplacian, Steklov problem, critical point theorem.

Mathematics Subject Classification: 35J48, 35J66.

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  • Mostafa Allaoui
  • University Mohamed I, Faculty of Sciences, Department of Mathematics, Oujda, Morocco
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2014-07-26.
  • Revised: 2014-11-10.
  • Accepted: 2014-11-13.
  • Published online: 2015-06-06.
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Cite this article as:
Mostafa Allaoui, Continuous spectrum of Steklov nonhomogeneous elliptic problem, Opuscula Math. 35, no. 6 (2015), 853-866, http://dx.doi.org/10.7494/OpMath.2015.35.6.853

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