Ground state solution of semilinear Schrödinger system with local super-quadratic conditions - Jing Chen; Yiqing Li (Electronic Journal of Qualitative Theory of Differential Equations, 2021)
In this paper, we dedicate to studying the following semilinear Schrödinger system \begin{equation*} \begin{cases} -\Delta u+V_1(x)u =F_{u}(x,u,v)&\mbox{in}~\mathbb{R}^N, \\ -\Delta v+V_2(x)v=F_{v}(x,u,v)&\mbox{in}~\mathbb{R}^N, \\ u, \ v\in H^1(\mathbb{R}^N), \end{cases} \end{equation*} where the potential $V_i$ are periodic in $x, i=1,2$, the nonlinearity $F$ is allowed super-quadratic at some $x\in\mathbb{R}^N$ and asymptotically quadratic at the other $x\in\mathbb{R}^N$. Under a local super-quadratic condition of $F$, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.