Ground State Solutions for Resonant Cooperative Elliptic Systems with General Superlinear Terms

Abstract

This paper is concerned with the following cooperative elliptic system: $$\left\{ \begin{array}{ll} -\Delta u=\xi u+f(x,u,v), \quad \quad \mbox{in } \Omega, -\Delta v=\zeta v+g(x,u,v), \quad \quad \mbox{in } \Omega, u=v=0, \quad \quad\mbox{on } \partial\Omega, \end{array} \right.$$ where \({U=(u,v): \Omega\rightarrow \mathbb{R}^{2}}\), Ω is a bounded smooth domain in \({\mathbb{R}^{N}}\) and \({\xi,\zeta\in\mathbb{R}}\). We establish the existence of ground state solutions for this system using a much more direct approach to find a minimizing Cerami sequence for the energy functional outside the generalized Nehari manifold developed recently by Szulkin and Weth.