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

Abstract

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 $N\geq4$, $0<μ\leq4$, $\varepsilon>0$ is a small parameter, $Ω$ is a bounded domain in $\mathbb{R}^N$, and $2_μ^\ast=\frac{2N-μ}{N-2}$ 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 $\varepsilon$ small.