Blow-up behavior of prescribed mass minimizers for nonlinear Choquard equations with singular potentials

We study the constrained minimizing problem for the energy functional related to the nonlinear Choquard equation I(a) = inf {E(phi) : phi is an element of H-1(R-N), parallel to phi parallel to(2)(L2) = a}, where N >= 1, a > 0, E(phi):= 1/2 integral(RN) vertical bar del phi(x)vertical bar(2)dx+1/2 integral V-RN(x)vertical bar del phi(x)vertical bar(2)dx - 1/2p integral(RN) (Ia *vertical bar phi|(p))(x)vertical bar phi(x)vertical bar(p)dx is the energy functional with 0 < a < N, N+ a/N < p < N+ a/N-2 if N = 3, N+ a/N < p < infinity if N = 1, 2 and I-alpha is the Riesz potential. The external potential V : R-N -> R is assumed to satisfy (A1) V is an element of L-q(R-N) + L-infinity(R-N) with q = 1 if N = 1, q > 1 if N = 2 and q = N/2 if N >= 3 and (A2) for any c > 0, |{x is an element of R-N : vertical bar V(x)vertical bar > c}| < infinity. We first give a complete classification of existence and non-existence of minimizers for the problem. In the mass-critical case p = N+a+2/N, under an appropriate assumption of the external potential, we give a detailed description of the blow-up behavior of minimizers as the mass tends to a critical value.