Bound state positive solutions for a Hartree system with nonlinear couplings

In this paper, we are interested in the following Hartree system with nonlinear couplings: ............................................... - e2 u + V1 (x) u = 1 eN- mu .1 RN |u|p |x - y| mu dy |u|p-2u + ss RN |v|q |x - y| mu dy |u|q-2u, - e2 v + V2 (x) v = 1 eN- mu .2 RN |v|p |x - y|mu dy |v|p-2v + ss RN |u|q |x - y| mu dy |v|q-2v, u, v. H1(RN), u, v > 0 in RN, where N= 3, 0< mu< N, 2N- mu N = p= 2N- mu N-2, 2N- mu N = q = min{p, 2},.1,.2 > 0, e is a small parameter and ss < 0 is a coupling constant, and the potentials V1 and V2 have k1 and k2 isolated global minimum points, respectively. Using the Nehari manifold technique, the energy estimate method and the Lusternik-Schnirelmann theory, we find an interesting phenomenon that the problem possesses k1k2 positive solutions when V1 and V2 do not have any common isolated global minimum points, and k1k2 + m positive solutions when V1 and V2 have m common isolated global minimum points. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.