Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$-\mathrm{\Delta }u=(I_{\mu }\ast u^{2_{\mu }^{\ast }})u^{2_{\mu }^{\ast }-1},x\in {\mathbb{R}}^N,$$ where $0<\mu <N$, if $N=3\text{or}4$ and $0<\mu \le 4$ if $N\ge 5$, $I_{\mu }$ is the Riesz potential defined by $$I_{\mu }(x)=\frac{\mathrm{\Gamma }(\frac{\mu }{2})}{\mathrm{\Gamma }(\frac{N-\mu }{2}){\pi }^{\frac{N}{2}}{2}^{N-\mu }\mathrm{\mid }x{\mathrm{\mid }}^{\mu }}$$ with $\mathrm{\Gamma }(s)={\int }_0^{+\mathrm{\infty }}{x}^{s-1}{e}^{-x}dx$, $s>0$ and $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove the symmetry and uniqueness of the positive solutions and prove the nondegeneracy of the unique solutions for the equation when $\mu$ close to $N$.