Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications

2025/08/11 by Minbo Yang, Yang, Minbo, Shuijin Zhang +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Critical exponent #Dimension (graph theory) #Exponent #FOS: Mathematics #Geometric Analysis and Curvature Flows #Group (periodic table) #Heisenberg group #Homogeneous #Nonlinear Partial Differential Equations #Uniqueness

paper · pdf · doi:10.48550/arxiv.2508.07719

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the uniqueness and nondegeneracy of positive bubble solutions for the generalized energy-critical Hartree equation on the Heisenberg group ℍⁿ, -Δ_ℍu=(∫_ℍⁿ\frac|u(η)|^Q^∗_μ|η^-1ξ|^μdη)|u|^Q^∗_μ-2u,~~~ξ,η∈ℍⁿ, where Δ_ℍ represents the Kohn Laplacian, u(η) is a real-valued function, Q=2n+2 is the homogeneous dimension of ℍⁿ, μ∈ (0,Q) is a real parameter and Q^∗_μ is the upper critical exponent following the Hardy-Littlewood-Sobolev inequality on the Heisenberg group. By applying the Cayley transform, the spherical harmonic decomposition and the Funk-Hecke formula of the spherical harmonic function, we prove the nondegeneracy of positive bubble solutions for (\ref0.1). As an applications, we investigate the asymptotic behavior of the solutions for the Brezis-Nirenberg type problem as ε→ 0 \ \beginaligned amp;-Δ_ℍu=ε u+(∫_Ω\frac|u(η)|^Q^∗_μ|η^-1ξ|^μdη)|u|^Q^∗_μ-2u,~~amp;amp;in~Ω⊂ ℍⁿ, amp;u=0,~~amp;amp;on~∂Ω. \endaligned .

Citations

• Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
• Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations
• Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type
• Improved higher-order Sobolev inequalities on CR sphere
• Sharp quantitative estimates of Struwe's Decomposition
• Sharp gradient stability for the Sobolev inequality
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation
• Remainder terms in the fractional Sobolev inequality
• Sharp constants in several inequalities on the Heisenberg group

Related

• Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations
• Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group
• Nondegeneracy of solutions for a critical Hartree equation
• Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in \R⁶
• A Liouville theorem in the Heisenberg group