Elliptic $p$-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -\Delta_{p_1}u_1 & =-|u_1|^{p_1-2}u_1 \quad && \text{in } \Omega,\newline -\Delta_{p_2}u_2 & =-|u_2|^{p_2-2}u_2 \quad && \text{in } \Omega,\newline |\nabla u_1|^{p_1-2}\nabla u_1 \cdot \nu &=g_1(x,u_1,u_2) && \text{on } \partial\Omega,\newline |\nabla u_2|^{p_2-2}\nabla u_2 \cdot \nu &=g_2(x,u_1,u_2) && \text{on } \partial\Omega, \end{aligned} \end{equation*} where $\nu(x)$ is the outer unit normal of $\Omega$ at $x \in \partial\Omega$, $\Delta_{p_i}$ denotes the $p_i$-Laplacian and $g_i\colon \partial\Omega \times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory functions that satisfy general growth and structure conditions for $i=1,2$. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of $g_i$ near zero related to the first eigenvalue of the $p_i$-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, $(g_1,g_2)=\nabla g$ with a smooth function $(s_1,s_2)\mapsto g(x,s_1,s_2)$. By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the $p_i$-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.