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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