Quantitative stability of critical points for the nonlocal-Sobolev inequality in Heisenberg group

2025/08/12 by Shuijin Zhang, Zhang, Shuijin, Jijie Xu +3
Mathematics · #Nonlinear Partial Differential Equations #Differential Equations and Boundary Problems #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2508.08614

Abstract

We investigate the quantitative stability of the nonlocal Sobolev inequality in Heisenberg group C_HL(Q,μ) (∫_ℍⁿ∫_ℍⁿ\frac|u(ξ)|^Q^∗_μ|u(η)|^Q^∗_μ|η^-1ξ|^μdξdη)^\frac1Q^∗_μ≤ ∫_ℍⁿ|∇_Hu|²dξ, ∀ u∈ S^1,2(ℍⁿ), where Q=2n+2 is the homogeneous dimension of the Hiesenberg group ℍⁿ, μ∈(0,Q) and Q^∗_μ=(2Q-μ)/(Q-2) are two parameters corresponding to the Hardy-Littlewood-Sobolev inequality and Folland-Stein inequality on Heisenberg group, C_HL(Q,μ) is the sharp constant of the nonlocal-Sobolev inequality. Specifically, when u is close to solving the Euler equation -Δ_H u=(∫_ℍⁿ\frac|u(η)|^Q^∗_μ|η^-1ξ|^μdη)|u|^Q^∗_μ-2u, ξ,η∈ℍⁿ, the natural distance between u and the the set of optimizers U_λ,ζ, defined as δ(u)=||∇_Hu-∇_HU_λ,ζ||_L², can be linearly bounded by the functional derivative term Γ(u)=‖Δ_Hu+(∫_ℍⁿ\frac|u(η)|^Q^∗_μ|η^-1ξ|^μdη)|u|^Q^∗_μ-2u‖_{(S^1,2(ℍⁿ))^-1}. And for the weakly interacting bubble solutions \mathop∑_i=1^νU_λ_i,ζ_i, the aforementioned quantitative stability result holds when the dimension Q=4.

Citations

• Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants
• Symmetry and classification of solutions to an integral equation in the Heisenberg group ℍⁿ
• Dynamics of radial threshold solutions for generalized energy-critical Hartree equation
• Remainder terms of a nonlocal Sobolev inequality1
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• Stability for the Sobolev inequality: existence of a minimizer
• Sharp quantitative estimates of Struwe's Decomposition
• Improved Sobolev inequality under constraints
• Sharp gradient stability for the Sobolev inequality
• Uniqueness and nondegeneracy of solutions for a critical nonlocal equation
• On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation
• Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
• Remainder terms in the fractional Sobolev inequality
• Sharp constants in several inequalities on the Heisenberg group
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Related

• Stability estimates for critical points of a nonlocal Sobolev-type inequality
• Quantitative stability of a nonlocal Sobolev inequality
• Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants
• Remainder terms, profile decomposition and sharp quantitative stability in the fractional nonlocal Sobolev-type inequality with n>2s
• Sharp constants in several inequalities on the Heisenberg group