Classification of solutions of higher order critical Choquard equation

Abstract

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