Symmetry and classification of solutions to an integral equation in the Heisenberg group ${\mathbb{H}}^n$

In this paper we prove symmetry of nonnegative solutions of the integral equation $$u(\zeta )=\underset{{\mathbb{H}}^n}{\int }\mathrm{\mid }{\zeta }^{-1}\xi {\mathrm{\mid }}^{-(Q-\alpha )}u(\xi {)}^{p}d\xi 1<p\le \frac{Q+\alpha }{Q-\alpha },0<\alpha <Q$$ on the Heisenberg group ${\mathbb{H}}^n={\mathbb{C}}^n\times \mathbb{R}$, $Q=2n+2$ using the moving plane method and the Hardy-Littlewood-Sobolev inequality proved by Frank and Lieb for the Heisenberg group. For $p$ subcritical, i.e., $1<p<\frac{Q+\alpha }{Q-\alpha }$ we show nonexistence of positive solution of this integral equation, while for the critical case, $p=\frac{Q+\alpha }{Q-\alpha }$ we prove that the solutions are cylindrical and are unique upto Heisenberg translation and suitable scaling of the function $$u_0(z,t)={((1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^2+t^2)}^{-\frac{Q-\alpha }{4}}(z,t)\in {\mathbb{H}}^n\mathrm{.}$$ As a consequence, we also obtain the symmetry and classification of nonnegative $C^2$ solution of the equation $${\mathrm{\Delta }}_{\mathbb{H}}u+u^p=0for1<p\le \frac{Q+\alpha }{Q-\alpha }in{\mathbb{H}}^n$$ without any partial symmetry assumption on the function $u$.