Uniqueness of positive solutions of the Choquard type equations

In this paper, by developing Lieb's methods, we establish uniqueness of the positive radial solutions to the following solitary Choquard type equation -Delta phi + phi = (integral(RN) vertical bar phi(y)vertical bar(2)/vertical bar x - y vertical bar N-2 dy) phi, phi is an element of H-1 (R-N) This, together with Ma and Zhao's classification results, enables us to show the uniqueness of the positive solutions when N = 3, 4, 5.