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.