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Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group

Shuijin Zhang, Jialin Wang, Yu Zheng, Xiang Li, Jijie Xu

TL;DR

This work classifies positive solutions of the critical Hartree equation on the Heisenberg group by adapting the moving plane method to the noncommutative, sub-elliptic setting. Key steps include establishing invariance under group operations and CR-type inversion, and then applying the moving plane method in integral form to prove cylindrical symmetry up to translations and dilations. The authors further show that solutions possess CR inversion symmetry with respect to the unit CC-sphere, which, together with a decay analysis, yields that every positive solution is the CR inversion of the standard profile $u_{0}( ho) = ((1+|z|^{2})^{2}+t^{2})^{- rac{Q-2}{4}}$ up to translation and scaling. This combination of symmetry and inversion arguments leads to the uniqueness of the positive solution, providing a complete rigidity result for the nonlocal Hartree equation on $\mathbb{H}^{n}$.

Abstract

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}).

Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group

TL;DR

This work classifies positive solutions of the critical Hartree equation on the Heisenberg group by adapting the moving plane method to the noncommutative, sub-elliptic setting. Key steps include establishing invariance under group operations and CR-type inversion, and then applying the moving plane method in integral form to prove cylindrical symmetry up to translations and dilations. The authors further show that solutions possess CR inversion symmetry with respect to the unit CC-sphere, which, together with a decay analysis, yields that every positive solution is the CR inversion of the standard profile up to translation and scaling. This combination of symmetry and inversion arguments leads to the uniqueness of the positive solution, providing a complete rigidity result for the nonlocal Hartree equation on .

Abstract

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 denotes the Kohn Laplacian, is a real-valued function, is the homogeneous dimension of , is a real parameter and is the upper critical exponent associated with the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By introducing the -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}).

Paper Structure

This paper contains 8 sections, 12 theorems, 151 equations.

Table of Contents

  1. Introduction and Main Results
  2. Classification of positive solutions for semilinear elliptic equations on $\mathbb{R}^{N}$
  3. Classification of positive solutions for sub-elliptic equations on $\mathbb{H}^{n}$
  4. Preliminary Results
  5. Invariance under group operations
  6. Invariance under CR type iversion
  7. Moving plane method in integral form
  8. Uniqueness of the Positive Solution for the Nonlocal Equation

Key Result

Theorem 1.1

Assume that $Q\geq 4$, $0<\mu\leq4$ and $Q^{\ast}_{\mu}=\frac{2Q-\mu}{Q-2}$. Let $(u,v)\in L^{Q^{\ast}}_{loc}(\mathbb{H}^{n})\times L^{\frac{2Q}{\mu}}_{loc}(\mathbb{H}^{n})$ be a pair of positive solution of system (system). Then (i) There exists $\zeta_{0}\in\mathbb{H}^{n}$ such that $u$ is $\mathb Particular, $u$ and $v$ must be cylindrical symmetry up to a Heisenberg translation; (ii) The limit

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 13 more