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 {-Δu=(∫Ωu2μ∗(ξ)|x-ξ|μdξ)u2μ∗-1+εu,inΩ,u=0,on∂Ω, where N≥ 4 , 0 < μ≤ 4 , ε> 0 is a small parameter, Ω is a bounded domain in RN , and 2μ∗=2N-μN-2 is the critical exponent in the sense of the Hardy–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 ε small.