Isolated singularities of solutions of defocusing Hartree equation

Our purpose of this paper is to study isolated singular positive solution u of defocusing Hartree equation (1){−Δu+u+Iα[up]uq=0inRN∖{0},lim|x|→+∞u(x)=0, where p,q>0, N≥3, α∈(0,N) and Iα[up](x)=∫RNu(y)p|x−y|N−αdy. We obtain that the solution u∈Lloc1(RN) of (1) satisfying that up∈L1(RN), Iα[up]uq∈Lloc1(RN), is a weak solution of (2){−Δu+u+Iα[up]uq=kδ0inRN,lim|x|→+∞u(x)=0. Furthermore, the classical solution of (1) is derived by considering the very weak solution of (2). To this end, we make use of Schauder fixed point theorem to obtain the existence of weak solutions of (2).