Electron. J. Diff. Equ., Vol. 2012 (2012), No. 132, pp. 1-12.

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

Mostafa Allaoui, Abdel Rachid El Amrouss, Anass Ourraoui

Abstract:
In this article we study the nonlinear Steklov boundary-value problem
$$\displaylines{ \Delta_{p(x)} u=|u|^{p(x)-2}u \quad \hbox{in } \Omega, \cr |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda f(x,u) \quad \hbox{on } \partial\Omega. }$$
Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.

Submitted February 24, 2012. Published August 15, 2012.
Math Subject Classifications: 35J48, 35J60, 35J66
Key Words: p(x)-Laplace operator; variable exponent Lebesgue space; variable exponent Sobolev space; Ricceri's variational principle.

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Mostafa Allaoui
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: allaoui19@hotmail.com
Abdel Rachid El Amrouss
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: elamrouss@hotmail.com
Anass Ourraoui
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: anas.our@hotmail.com

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