Solutions for Steklov boundary value problems involving p(x)-Laplace operators

Abstract

In this paper we study the nonlinear Steklov boundary value problem of the following form: $$ (\mathcal{S}) \left\{ \begin{array}{lr} ~~\Delta_{p(x)} u=|u|^{p(x)-2}u & \mbox{in}~~ \Omega , \\ ~~~|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda f(x,u) & \mbox{on}~ \partial\Omega . \end{array} \right. $$ Using the variational method, under appropriate assumptions on $f$, we establish the existence of at least three solutions of this problem.

Keywords

• p(x)-Laplace operator
• Embedding theorem
• variable exponent
• Sobolev space
• Ricceri´s variational principle