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 \begin{equation*} \begin{cases} -Δu =\displaystyle{\Big(\int_Ω\frac{u^{2_μ^\ast} (ξ)}{|x-ξ|^μ}dξ\Big)u^{2_μ^\ast-1}}+\varepsilon u ,~$\text{in}$~Ω,\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$\text{on}$~\partialΩ, \end{cases} \end{equation*} where $N\ge 4$, $0<\mu \le 4$, $\epsilon >0$ is a small parameter, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$, and $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$ 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 $\epsilon$ small.