Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials
Abstract
Let n≥2, 0<β<n, and let V be a continuous nonnegative potential on Rn with a positive lower bound outside a ball. Under the constraint ∫Rn(|∇ u|n+V|u|n)dd≤1, we prove the singular Moser--Trudinger inequality with sharp coefficient (1−β/n)αn, where αn=n|𝕊n−1|1/(n−1). The subcritical supremum is attained for every such potential. If V is radial and rβV(r) is nondecreasing, the critical supremum is also attained. This includes potentials that vanish on a ball and decrease to a positive limit at infinity. Under the same radial hypothesis, a bound for the tangential gradient shows that every extremal is radial and that the tangential gradients of maximizing sequences converge to zero in Ln, including at the critical coefficient. For the singular n-Laplacian equation with critical exponential growth, we obtain an explicit sufficient level for Palais--Smale compactness. Under the Ambrosetti--Rabinowitz condition and a smallness assumption at zero, this gives a nonzero nonnegative mountain-pass solution for all sufficiently large coupling parameters.
Related
- On sharp singular Moser-Trudinger type inequalities and applications to zero mass (p,Q)-Laplace subelliptic equations in the Heisenberg group
- Sobolev Inequalities and the Existence of Solutions to Degenerate p-Poisson Equations
- Observability Inequalities for Degenerate Schrödinger Equations via a Shape Approximation Method
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
- Discrete Sobolev and Trudinger inequalities on polyhedral meshes
BibTeX
@misc{hu2026sharp,
title = {{Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials}},
author = {Hao Hu and Yanjun Liu and Huabo Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07628},
eprint = {2611.07628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07628},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials
AU - Hao Hu
AU - Yanjun Liu
AU - Huabo Zhang
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07628
UR - https://arxiv.org/abs/2611.07628
AB - Let n≥2, 0<β<n, and let V be a continuous nonnegative potential on R^n with a positive lower bound outside a ball. Under the constraint ∫_R^n(|∇ u|^n+V|u|^n)dd≤1, we prove the singular Moser--Trudinger inequality with sharp coefficient (1−β/n)α_n, where α_n=n|𝕊^n−1|^1/(n−1). The subcritical supremum is attained for every such potential. If V is radial and r^βV(r) is nondecreasing, the critical supremum is also attained. This includes potentials that vanish on a ball and decrease to a positive limit at infinity. Under the same radial hypothesis, a bound for the tangential gradient shows that every extremal is radial and that the tangential gradients of maximizing sequences converge to zero in L^n, including at the critical coefficient. For the singular n-Laplacian equation with critical exponential growth, we obtain an explicit sufficient level for Palais--Smale compactness. Under the Ambrosetti--Rabinowitz condition and a smallness assumption at zero, this gives a nonzero nonnegative mountain-pass solution for all sufficiently large coupling parameters.
ER -