Classification of isolated singularities of positive solutions for Choquard equations

In this paper we classify the isolated singularities of positive solutions to Choquard equation−Δu+u=Iα[up]uqinRN∖{0},lim|x|→+∞⁡u(x)=0, where p>0,q≥1,N≥3,α∈(0,N)p>0,q≥1,N≥3,α∈(0,N) and Iα[up](x)=∫RNu(y)p|x−y|N−αdy. We show that any positive solution u is a solution of−Δu+u=Iα[up]uq+kδ0inRN in the distributional sense for some k≥0k≥0, where δ0δ0 is the Dirac mass at the origin. We prove the existence of singular solutions in the subcritical case: p+q