Abstract: We are concerned with the important system of nonlinear Schrodinger equations with linear and nonlinear couplings which arises from Bose-Einstein condensates, we prove phase segregation results of the limit competition case, we use variational methods to prove the existence of ground state and bound state solutions of the systems, and use bifurcation theory to get structure of positive solutions. We give some partial symmetry results of positive solutions by Morse index and obtain existence and uniqueness of positive solution via synchronized solution techniques etc.
