Existence of bound states for schrödinger-newton type systems

In this paper we consider the following elliptic system in R3 {equation presented} are non-negative real functions defined on R3 so that lim {equation presented} and lim {equation presented} When K(x) ≡ K∞ and a(x) ≡ a∞ we have already proved the existence of a radial ground state of the above system. Here, by using a new version of the moving plane method, we show that all positive solutions of the above system with K(x) ≡ K∞ and a(x) ≡ a∞ are radially symmetric and the linearized operator around a radial ground state is also non-degenerate. Using these results we further prove, under additional assumptions on K(x) and a(x), but not requiring any symmetry property on them, the existence of a positive solution for the system.