Classification of solutions of higher order critical Choquard equation

In this paper, we classify the solutions of the following critical Choquard equation $$(-\mathrm{\Delta }{)}^{\frac{n}{2}}u(x)={\int }_{{\mathbb{R}}^n}\frac{e^{\frac{2n-\mu }{2}u(y)}}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}dy{e}^{\frac{2n-\mu }{2}u(x)},\text{ in }{\mathbb{R}}^n,$$ where $0<\mu <n$, $n\ge 2$. Suppose $u(x)=o(\mathrm{\mid }x{\mathrm{\mid }}^2)\text{ at }\mathrm{\infty }$ for $n\ge 3$ and satisfies $${\int }_{{\mathbb{R}}^n}{e}^{\frac{2n-\mu }{2}u(y)}dy<\mathrm{\infty },{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{e^{\frac{2n-\mu }{2}u(y)}}{\mathrm{\mid }x-y{\mathrm{\mid }}^{\mu }}{e}^{\frac{2n-\mu }{2}u(x)}dydx<\mathrm{\infty }\mathrm{.}$$ By using the method of moving spheres, we show that the solutions have the following form $$u(x)=\ln \frac{C_1(\epsilon )}{\mathrm{\mid }x-x_0{\mathrm{\mid }}^2+{\epsilon }^2}\mathrm{.}$$