Ground state solution of semilinear Schrödinger system with local super-quadratic conditions

In this paper, we dedicate to studying the following semilinearSchrödinger system e q u a t i o n *- Δ u + V 1 ( x ) u = Fu( x , u , v )a m p ;m b o x i n ~RN ,r - Δ v + V 2 ( x ) v = Fv( x , u , v )a m p ;m b o x i n ~RN ,r u ,v ∈ H 1 (RN ) ,e n d e q u a t i o n *where the potential V i are periodic in x , i = 1 , 2 , the nonlinearity F is allowed super-quadraticat some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F , anapproximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energysolutions.