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

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, where, Vμ, 1/2: N is an external potential defined for μ, 1/2 > 0 and x N by Vμ, 1/2(x) = 1 -μ/( 1/22 + |x|2) and is the Riesz potential for α (0, N), we exhibit two thresholds μ1/2, μ1/2 > 0 such that the equation admits a positive ground state solution if and only if μ1/2 < μ < μ1/2 and no ground state solution exists for μ < μ1/2. Moreover, if μ > maxμ1/2, N2(N -2)/4(N + 1), then equation still admits a sign changing ground state solution provided or in dimension N = 3 if in addition 3/2 < α < 3 and, namely in the non-resonant case