Multiple solutions of a coupled nonlinear Schrödinger system

We will consider the relation between the number of positive standing waves solutions for a class of coupled nonlinear Schrödinger system in RN and the topology of the set of minimum points of potential V(x). The main characteristics of the system are that its functional is strongly indefinite at zero and there is a lack of compactness in RN. Combining the dual variational method with the Nehari technique and using the Concentration–Compactness Lemma, we obtain the existence of multiple solutions associated to the set of global minimum points of the potential V(x) for ϵ sufficiently small. In addition, our result gives a partial answer to a problem raised by Sirakov about existence of solutions of the perturbed system.