Nondegeneracy of solutions for a critical Hartree equation

The aim of this paper is to prove the nondegeneracy of the unique positive solutions for the following critical Hartree type equations when $μ>0$ is close to $0$, $-Δu=\left(I_μ\ast u^{2^{\ast}_μ}\right)u^{{2}^{\ast}_μ-1},~~x\in\mathbb{R}^{N},$ where $I_μ(x)=\frac{Γ(\fracμ{2})}{Γ(\frac{N-μ}{2})π^{\frac{N}{2}}2^{N-μ}|x|^μ}$ is the Riesz potential and $2^{\ast}_μ=\frac{2{N-μ}}{N-2}$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality.

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