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

2025/11/25 by Shuijin Zhang, Zhang, Shuijin, Jialin Wang +7
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2511.20264

openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28

Abstract

We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group -Δ_ℍu=(∫_ℍⁿ\frac|u(ξ)|^Q^∗_μ|ζ^-1ξ|^μdξ)|u|^Q^∗_μ-2u,~~~ζ,ξ∈ℍⁿ, where Δ_ℍ denotes the Kohn Laplacian, u(ξ) is a real-valued function, Q=2n+2 is the homogeneous dimension of ℍⁿ, μ∈ (0,Q) is a real parameter and Q^∗_μ=(2Q-μ)/(Q-2) is the upper critical exponent associated with the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By introducing the ℍ-reflection, we prove that the solutions of (\ref0.1) are cylindrical, upto Heisenberg translation and suitable scaling of function u₀(ζ)=u₀(z,t)=((1+|z|²)²+t²)^-(Q-2)/(4),~~~ζ=(z,t)∈ ℍⁿ. 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 (\ref0.1).

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