Local uniqueness of blow-up solutions for critical Hartree equations in bounded domain

In this paper we are interested in the following critical Hartree equation {(u=0) -Delta u = (integral(Omega) u(2)mu*(xi)/vertical bar x-xi vertical bar(mu) d xi)u(2)mu*(-1) = epsilon u, in Omega, on partial derivative Omega, where N >= 4, 0 < mu <= 4, epsilon > 0 is a small parameter, Omega is a bounded domain in R-N, and 2 mu* = 2N- mu/N-2 is the critical exponent in the sense of theHardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for epsilon small.

Automated summary

This paper investigates the critical Hartree equation with various methods and establishes the existence of blow-up solutions. It focuses on the study of the blow-up points for single bubbling solutions and proves the local uniqueness of blow-up solutions concentrated at non-degenerate critical points.