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=I_α[u^p]u^qinR^N∖{0},lim|x|→+∞⁡u(x)=0, where p>0,q≥1,N≥3,α∈(0,N) and I_α[u^p](x)=∫_R^N u(y)p|x−y|N−αdy. We show that any positive solution u is a solution of−Δu+u=I_α[u^p]u^q+kδ₀inR^N in the distributional sense for some k≥0, where δ₀ is the Dirac mass at the origin. We prove the existence of singular solutions in the subcritical case: p+q<N+αN−2andp<NN−2,q<NN−2 and prove that either the solution u has removable singularity at the origin or satisfies lim_|x|→0⁺ ⁡u(x)|x|^N−2=C_N which is a positive constant. In the supercritical case: p+q≥N+αN−2orp≥NN−2,orq≥NN−2 we prove that k=0.