Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities
Abstract
We combine spectral perturbation with orthogonal transformations in H01(Ω) to study singular Moser-Trudinger inequalities under the energy constraint ∫Ω|∇ u|2 dx≤1, where Ω is a smooth bounded planar domain containing the origin. For the weight |x|−2β, 0≤β<1, the positive maximizer is unique for small α>0, and the sharp global stability constant satisfies Cα=α/2 (1/μ1−1/μ2)+O(α2), α↓0, where μ1<μ2 are the first two weighted Dirichlet eigenvalues. We compute the second asymptotic coefficient by a weighted quartic maximization on the second eigenspace, even when the second eigenvalue is multiple. An exact reduction to the sphere orthogonal to the maximizer identifies the attaining functions and yields a limiting global-to-local ratio of 1/2. On the disk, the energy derivative with respect to the central value characterizes radial constrained degeneracy and yields finitely many exceptional normalized parameters. Combined with known critical uniqueness and nondegeneracy, this gives uniform quadratic stability near criticality, while compactness and analytic reduction yield finite-power stability throughout the admissible range.
Related
- Extremals for Trudinger-Moser inequalities via polygonal profiles
- Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities
- A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals
- Best Constants for Anisotropic Moser-Trudinger Inequalities with the Exact Growth Condition and Their Applications
- Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
BibTeX
@misc{guo2026unique,
title = {{Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities}},
author = {Kaiwen Guo and Yanjun Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38592},
eprint = {2610.38592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38592},
}
RIS
TY - UNPB
TI - Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities
AU - Kaiwen Guo
AU - Yanjun Liu
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38592
UR - https://arxiv.org/abs/2610.38592
AB - We combine spectral perturbation with orthogonal transformations in H_0^1(Ω) to study singular Moser-Trudinger inequalities under the energy constraint ∫_Ω|∇ u|^2 dx≤1, where Ω is a smooth bounded planar domain containing the origin. For the weight |x|^−2β, 0≤β<1, the positive maximizer is unique for small α>0, and the sharp global stability constant satisfies C_α=α/2 (1/μ_1−1/μ_2)+O(α^2), α↓0, where μ_1<μ_2 are the first two weighted Dirichlet eigenvalues. We compute the second asymptotic coefficient by a weighted quartic maximization on the second eigenspace, even when the second eigenvalue is multiple. An exact reduction to the sphere orthogonal to the maximizer identifies the attaining functions and yields a limiting global-to-local ratio of 1/2. On the disk, the energy derivative with respect to the central value characterizes radial constrained degeneracy and yields finitely many exceptional normalized parameters. Combined with known critical uniqueness and nondegeneracy, this gives uniform quadratic stability near criticality, while compactness and analytic reduction yield finite-power stability throughout the admissible range.
ER -