Solutions on asymptotically periodic elliptic system with new conditions (Q346591)

From the abstract: This paper is concerned with the following elliptic system: \[ \begin{cases} -\Delta u+U_{1}(x)u =F_{u}(x,u,v)\quad & \text{in }\mathbb{R}^{N},\\ -\Delta v+U_{2}(x)v=F_{v}(x,u,v) \quad & \text{in }\mathbb{R}^{N},\\ u, \;v \in H^{1}(\mathbb{R}^{N}). \end{cases} \] Assuming that the potential \(U_i(x)\) are periodic in \(x\) and \(0\) lies in a spectral gap of \(\sigma(-\Delta + U_i)\), \(i=1,2\), two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity \(F\) which are weaker that some well known ones. For the case that \(U_i(x)\) and \(F(x,u,v)\) are asymptotically periodic in \(x\), a nontrivial solution is established by using a generalized linking theorem and some new techniques.