We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group \begin{ equation}\label{ 0.1} -\Delta_{ \mathbb{ H}}u=\left(\int_{ \mathbb{ H}^{ n}}\frac{ |u(\xi)|^{ Q^{ \ast}_{ \mu}}}{ |\zeta^{ -1}\xi|^{ \mu}}\mathrm{ d}\xi\right)|u|^{ Q^{ \ast}_{ \mu}-2}u,~~~\zeta,\xi\in\mathbb{ H}^{ n}, \end{ equation} where $\Delta_{ \mathbb{ H}}$ denotes the Kohn Laplacian, $u(\xi)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{ H}^{ n}$, $\mu\in (0,Q)$ is a real parameter and $Q^{ \ast}_{ \mu}=\frac{ 2Q-\mu}{ Q-2}$ is the upper critical exponent associated with the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By introducing the $\mathbb{ H}$-reflection, we prove that the solutions of (\ref{ 0.1}) are cylindrical, upto Heisenberg translation and suitable scaling of function \begin{ equation*}\label{ 0.2} u_{ 0}(\zeta)=u_{ 0}(z,t)=\left((1+|z|^{ 2})^{ 2}+t^{ 2}\right)^{ -\frac{ Q-2}{ 4}},~~~\zeta=(z,t)\in \mathbb{ H}^{ n}. \end{ equation*} Furthermore, we show that these positive solutions are also CR inversion-symmetric with respect to the unit CC sphere. Consequently, we establish the uniqueness of positive solutions to equation (\ref{ 0.1}).
Categories: math.AP