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

Abstract

In this paper we prove symmetry of nonnegative solutions of the integral equation \[ u (\zeta ) = \int\limits_{{\mathbb H}^n} |\zeta^{-1} \xi|^{-(Q-\alpha)} u(\xi)^{p} d\xi \quad 1< p \leq \frac{Q+\alpha}{Q-\alpha},\quad 0< \alpha <Q \] 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 \[ u_0 (z,t) = \left( (1+ |z|^2)^2 + t^2 \right)^{- \frac{Q-\alpha}{4}} \quad (z,t ) \in {\mathbb H}^n. \] As a consequence, we also obtain the symmetry and classification of nonnegative $C^2$ solution of the equation \[ \Delta_{\mathbb H} u + u^{p} = 0 \quad \mbox{for } 1< p \leq \frac{Q+\alpha}{Q-\alpha} \mbox{ in } {\mathbb H}^n \] without any partial symmetry assumption on the function $u$.