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

Abstract

AbstractIn this paper we are interested in the following critical Hartree equation $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u =\displaystyle {\Big (\int _{\Omega }\frac{u^{2_{\mu }^*} (\xi )}{|x-\xi |^{\mu }}d\xi \Big )u^{2_{\mu }^*-1}}+\varepsilon u,~~~\text {in}~\Omega ,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text {on}~\partial \Omega , \end{array}\right. } \end{aligned}$$ - Δ u = ( ∫ Ω u 2 μ ∗ ( ξ ) | x - ξ | μ d ξ ) u 2 μ ∗ - 1 + ε u , in Ω , u = 0 , on ∂ Ω , where $$N\ge 4$$ N ≥ 4 , $$0<\mu \le 4$$ 0 < μ ≤ 4 , $$\varepsilon >0$$ ε > 0 is a small parameter, $$\Omega $$ Ω is a bounded domain in $$\mathbb {R}^N$$ R N , and $$2_{\mu }^*=\frac{2N-\mu }{N-2}$$ 2 μ ∗ = 2 N - 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 $$\varepsilon $$ ε small.