Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent

For the Choquard equation, which is a nonlocal nonlinear Schrodinger type equation, -.u + V mu,.u = (Ia * |u|(N+a)/N)|u|a/N- 1 u, in RN, where N3, V mu,. : RN. R is an external potential defined for mu,. > 0 and x. RN by V mu,. (x) = 1 - mu/(.2 + |x|2) and Ia : RN. 0 is the Riesz potential for a. (0, N), we exhibit two thresholds mu., mu. > 0 such that the equation admits a positive ground state solution if and only if mu. < mu < mu. and no ground state solution exists for mu < mu.. Moreover, if mu > max{mu., N2(N - 2)/4( N + 1)}, then equation still admits a sign changing ground state solution provided N4 or in dimension N = 3 if in addition 3/2 < a < 3 and ker(-.+ V mu,.) = {0}, namely in the non-resonant case.