Classification of isolated singularities of positive solutions for Choquard equations

In this paper we classify the isolated singularities of positive solutions to Choquard equation -Delta u + u = I-alpha[u(p)]u(q) in R-N \ {0}, lim vertical bar x vertical bar ->+infinity u (x) = 0, where p > 0, q >= 1, N >= 3, alpha is an element of(0, N) and I-alpha[u(P)](x) = integral R-N u(y)(P)/vertical bar x-y vertical bar(N-alpha)dy.We show that any positive solution u is a solution of -Delta u + u = I-alpha[u(P)]u(q) + k delta(0) in R-N in the distributional sense for some k >= 0, where delta(0) is the Dirac mass at the origin. We prove the existence of singular solutions in the subcritical case: p + q < N+alpha/N-2 and p < N/N-2, q < N/N-2 and prove that either the solution u has removable singularity at the origin or satisfies lim vertical bar x vertical bar -> 0+ u(x)vertical bar x vertical bar(N-2) = C-N which is a positive constant. In the supercritical case: p + q >= N+alpha/N-2 or p >= N/N-2, or q >= N/N-2 we prove that k = 0. (C) 2016 Elsevier Inc. All rights reserved.