Solutions for a Class of Nonperiodic Superquadratic Hamiltonian Elliptic Systems Involving Gradient Terms

Abstract

In the present paper, we consider the following Hamiltonian elliptic system $\left(\mathrm{H}\mathrm{E}\mathrm{S}\right)$: $-\mathrm{\Delta }u+b\left(x\right)·\nabla u+V\left(x\right)u=H_v\left(x,u,v\right),x\in {\mathbb{R}}^N$, $-\mathrm{\Delta }v-b\left(x\right)·\nabla v+V\left(x\right)v=H_u\left(x,u,v\right),x\in {\mathbb{R}}^N.$ A new existence result of nontrivial solutions is obtained for the system $\left(\mathrm{H}\mathrm{E}\mathrm{S}\right)$ via variational methods for strongly indefinite problems, which generalizes some known results in the literatures.