Existence for an elliptic system with nonlinear boundary conditions via fixed-point methods

Abstract

In this paper we prove the existence of nonnegative nontrivial solutions of the system $$\left\{\begin{array}{rcll} \Delta u & = & u & \mbox{in } \Omega,\\ \Delta v & = & v, & \end{array}\right.$$ with nonlinear coupling through the boundary given by $$ \left\{\begin{array}{rcll} \frac{\partial u}{\partial n} & = & f(x,u,v) & \mbox{on } \partial \Omega, \\ \frac{\partial v}{\partial n} & = & g(x,u,v), \end{array}\right. $$ under suitable assumptions on the nonlinear terms $f$ and $g$. For the proof we use a fixed-point argument and the key ingredient is a Liouville type theorem for a system of Laplace equations with nonlinear coupling through the boundary of power type in the half space.