Nondegeneracy of solutions for a critical Hartree equation

Abstract: The aim of this paper is to prove the nondegeneracy of the unique positive solutions for the following critical Hartree type equations when $\mu>0$ is close to $0$, $$ -\Delta u=\left(I_{\mu}\ast u^{2^{\ast}{\mu}}\right)u^{{2}^{\ast}{\mu}-1},~~x\in\mathbb{R}^{N}, $$ where $ I_{\mu}(x)=\frac{\Gamma(\frac{\mu}{2})}{\Gamma(\frac{{N-\mu}}{2})\pi^{\frac{N}{2}}2^{{N-\mu}}|x|^{\mu}} $ is the Riesz potential and $2^{\ast}_{\mu}=\frac{2{N-\mu}}{N-2}$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality.