Existence and multiplicity of solutions for a $(p(x),q(x))$-Laplacian Steklov problem

Résumé

Using variational methods, we prove in different cases the existence and multiplicity of solutions for the following Steklov problem
\begin{equation*}
\left\{
\begin{array}{ll}
\triangle_{p(x)}u+\triangle_{q(x)}u=0 & \text{in } \Omega, \\
(|\nabla u|^{p(x)-2}+|\nabla u|^{q(x)-2})\frac{\partial u}{\partial\nu}+|u|^{p(x)-2}u+|u|^{q(x)-2}u=\lambda( |u|^{r(x)-2}u-\varepsilon|u|^{s(x)-2}u ) & \text{on } \partial\Omega,
\end{array} \right.
\end{equation*}
where $\Omega\subset\mathbb{R}^N(N \geq 2)$ is a bounded domain with smooth boundary $\partial\Omega$ and $\nu$ is the unit outward normal vector on $\partial\Omega$. $p, \; q, \; r,\; s: \overline{\Omega} \mapsto (1,+\infty)$ are continuous functions and $\varepsilon \geq 0$.