Existence and non-existence of solutions for a (p,q)-Laplacian Steklov system

Abstract

In this paper, we study the existence and non-existence of a weak solutions to the following system:
$$\left\{
\begin{array}{ll}
\Delta_p u=\Delta_q v=0& \mbox{ in }\Omega\\
|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda m|u|^{p-2} u-\varepsilon[(\alpha+1)|u|^{\alpha-1}u |v|^{\beta+1}-f] & \mbox{ on }\partial\Omega\\
|\nabla v|^{q-2}\frac{\partial v}{\partial \nu}=\lambda n|v|^{q-2} v-\varepsilon[(\beta+1)|v|^{\beta-1}v |u|^{\alpha+1}-g] \mbox{ on } \partial\Omega,
\end{array}
\right.$$
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq2)$ with a smooth boundary $\partial\Omega$, $\Delta_pu=\mbox{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian, $\frac{\partial}{\partial\nu}$ is the outer normal derivative, $\varepsilon\in\{0, 1\}$, $ m, n$, $f$ and $g$ are functions that satisfy some conditions.