Super-quadratic conditions for periodic elliptic system on R^N

Abstract

This article concerns the elliptic system $$\displaylines{ -\Delta u+V(x)u=W_{v}(x, u, v), \quad x\in \mathbb{R}^{N},\cr -\Delta v+V(x)v=W_{u}(x, u, v), \quad x\in \mathbb{R}^{N},\cr u, v\in H^{1}(\mathbb{R}^{N}), }$$ where V and W are periodic in x, and W(x,z) is super-linear in z=(u,v). We use a new technique to show that the above system has a nontrivial solution under concise super-quadratic conditions. These conditions show that the existence of a nontrivial solution depends mainly on the behavior of W(x,u,v) as $|u+v| \to 0$ and $|au+bv| \to \infty$ for some positive constants a,b.

Keywords

• Elliptic system
• super-quadratic
• nontrivial solution
• strongly indefinite functionals