Existence of infinitely many solutions for a Steklov problem involving the p(x)-Laplace operator

• Main Authors: Mostafa Allaoui, Abdel Rachid El Amrouss, Anass Ourraoui
• Format: Article
• Language: English
• Published: University of Szeged 2014-05-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: $p(x)$-laplace operator; infinitely many solutions; variable exponent sobolev space; ricceri's variational principle
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2148

In this article, we study the nonlinear Steklov boundary-value problem $$\begin{alignedat}{2} \Delta_{p(x)}u & =|u|^{p(x)-2}u \quad &&\text{in } \Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu} & = f(x,u) \quad &&\text{on } \partial\Omega. \end{alignedat}$$ We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B.\ Ricceri and the theory of the variable exponent Sobolev spaces.