Existence of multiple solutions to a class of nonlinear Schrِdinger 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.