Existence of multiple solutions to a class of nonlinear Schrödinger system with external sources terms

We study a class of nonlinear Schrödinger system with external sources terms as perturbations in order to obtain existence of multiple solutions, this system arises from Bose-Einstein condensates etc. As these external sources terms are positive functions and small in some sense, we use Nehari manifold to get the existence of a positive ground state solution and a positive bound state solution.