Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents

We consider the following nonlocal critical equation\begin{document}$\begin{equation} -\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1) \end{equation}$\end{document}where \begin{document}$ 0<\mu_1,\mu_2<N $\end{document} if \begin{document}$ N = 3 $\end{document} or \begin{document}$ 4 $\end{document} , and \begin{document}$ N-4\leq\mu_1,\mu_2<N $\end{document} if \begin{document}$ N\geq5 $\end{document} , \begin{document}$ 2_{\mu_{i}}^\ast: = \frac{N+\mu_i}{N-2}(i = 1,2) $\end{document} is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and \begin{document}$ I_{\mu_i} $\end{document} is the Riesz potential\begin{document}$ \begin{equation*} I_{\mu_i}(x) = \frac{\Gamma(\frac{N-\mu_i}{2})}{\Gamma(\frac{\mu_i}{2})\pi^{\frac{N}{2}}2^{\mu_i}|x|^{N-\mu_i}}, \; i = 1,2, \end{equation*} $\end{document}with \begin{document}$ \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx $\end{document} , \begin{document}$ s>0 $\end{document} . Firstly, we prove the existence of the solutions of the equation (1). We also establish integrability and \begin{document}$ C^\infty $\end{document} -regularity of solutions and obtain the explicit forms of positive solutions via the method of moving spheres in integral forms. Finally, we show that the nondegeneracy of the linearized equation of (1) at \begin{document}$ U_0,V_0 $\end{document} when \begin{document}$ \max\{\mu_1,\mu_2\}\rightarrow0 $\end{document} and \begin{document}$ \min\{\mu_1,\mu_2\}\rightarrow N $\end{document} , respectively.