Ground states for a coupled Schrödinger system with general nonlinearities

We study a coupled Schrödinger system with general nonlinearities. By using variational methods, we prove the existence and asymptotic behaviour of ground state solution for the system with periodic couplings. Moreover, we prove the existence and nonexistence of ground state solution for the system with non-periodic couplings via Nehari manifold method. Especially, the ground state solution with both nontrivial components is obtained, and the sign of nontrivial components is considered.