Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities
Abstract
For N≥2, 0<β<N and q>1, we study maximizers of the critical singular anisotropic Moser--Trudinger integral ∫ℝN (ΦN,q,β (λN(1−β/N)|u|N/(N−1)))/Fo(x)β dx, |F(∇ u)|Na+|u|qb≤ 1. where a>0, 0<b≤ N and ΦN,q,β is the integrable Taylor remainder. The supremum is attained for every a>0 when b<N. For b=N and 1<q<q+, the first retained Taylor term separates two regimes, with q-:=(N2(N−2))/((N−1)(N−β)), q+:=N2/(N−β). Attainment holds for every a>0 if 1<q<q-, while for q>1, q-≤ q<q+, there is a finite threshold ac>N and attainment holds if and only if 0<a≤ ac. An exact Euclidean reduction shows that the threshold is independent of the anisotropy. The proof combines critical--subcritical scaling with a nonlinear Green-function concentration bound. Radial flux and Pohozaev identities determine the first correction to the subcritical supremum, and refined Green tests give the strict comparison needed for attainment at the finite threshold, including in dimension two.
Related
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BibTeX
@misc{guo2026extremals,
title = {{Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities}},
author = {Kaiwen Guo and Yanjun Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21250},
eprint = {2609.21250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21250},
}
RIS
TY - UNPB
TI - Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities
AU - Kaiwen Guo
AU - Yanjun Liu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.21250
UR - https://arxiv.org/abs/2609.21250
AB - For N≥2, 0<β<N and q>1, we study maximizers of the critical singular anisotropic Moser--Trudinger integral ∫_ℝ^N (Φ_N,q,β (λ_N(1−β/N)|u|^N/(N−1)))/F^o(x)^β dx, |F(∇ u)|_N^a+|u|_q^b≤ 1. where a>0, 0<b≤ N and Φ_N,q,β is the integrable Taylor remainder. The supremum is attained for every a>0 when b<N. For b=N and 1<q<q_+, the first retained Taylor term separates two regimes, with q_-:=(N^2(N−2))/((N−1)(N−β)), q_+:=N^2/(N−β). Attainment holds for every a>0 if 1<q<q_-, while for q>1, q_-≤ q<q_+, there is a finite threshold a_c>N and attainment holds if and only if 0<a≤ a_c. An exact Euclidean reduction shows that the threshold is independent of the anisotropy. The proof combines critical--subcritical scaling with a nonlinear Green-function concentration bound. Radial flux and Pohozaev identities determine the first correction to the subcritical supremum, and refined Green tests give the strict comparison needed for attainment at the finite threshold, including in dimension two.
ER -