Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 127, pp. 1-11.
Title: Super-quadratic conditions for periodic elliptic system on $\mathbb{R}^N$
Authors: Fangfang Liao (Central South Univ., Changsha, China)
Xianhua Tang (Central South Univ., Changsha, China)
Jian Zhang (Central South Univ., Changsha, China)
Dongdong Qin (Central South Univ., Changsha, China)
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.
Submitted January 31, 2015. Published May 06, 2015.
Math Subject Classifications: 35J10, 35J20.
Key Words: Elliptic system; super-quadratic; nontrivial solution;
strongly indefinite functionals.