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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.20264 |
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Table of Contents:
- 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} -Δ_{\mathbb{H}}u=\left(\int_{\mathbb{H}^{n}}\frac{|u(ξ)|^{Q^{\ast}_μ}}{|ζ^{-1}ξ|^μ}\mathrm{d}ξ\right)|u|^{Q^{\ast}_μ-2}u,~~~ζ,ξ\in\mathbb{H}^{n}, \end{equation} where $Δ_{\mathbb{H}}$ denotes the Kohn Laplacian, $u(ξ)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\mathbb{H}^{n}$, $μ\in (0,Q)$ is a real parameter and $Q^{\ast}_μ=\frac{2Q-μ}{Q-2}$ 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}(ζ)=u_{0}(z,t)=\left((1+|z|^{2})^{2}+t^{2}\right)^{-\frac{Q-2}{4}},~~~ζ=(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}).