PreprintCC BY 4.0arXiv
A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains
Abstract
\noindent Let Om⊂R2 be a smooth bounded simply connected domain. We prove that all positive solutions of −Δu=θueu2, with θ>0 and zero Dirichlet boundary data, have Dirichlet energy bounded by a constant depending only on Om. Consequently, the Trudinger--Moser functional has no nonnegative constrained critical points at sufficiently large prescribed energies. The proof combines a weighted Pohozaev identity obtained after conformal change of variables, concentration analysis of normalized measures, and local radial comparison. The comparison hypotheses are obtained from local integral estimates, without an a priori energy bound. An estimate for the exterior Green potential then excludes sequences of unbounded energy.
Related
- Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains
- Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities
- Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities
- SLOPE DETERMINATION ON CONVEX DOMAINS: DIRICHLET DATA AND CRANDALL-PAZY DIRECTIONS
- On Simply Connected Domains Supporting an Unbounded Analytic Function with Bounded Derivative
BibTeX
@misc{hu2026priori,
title = {{A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains}},
author = {Hongtao Hu and Qiaoran Wu and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10086},
eprint = {2610.10086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10086},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains
AU - Hongtao Hu
AU - Qiaoran Wu
AU - Wenming Zou
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10086
UR - https://arxiv.org/abs/2610.10086
AB - \noindent Let Om⊂R^2 be a smooth bounded simply connected domain. We prove that all positive solutions of −Δu=θue^u^2, with θ>0 and zero Dirichlet boundary data, have Dirichlet energy bounded by a constant depending only on Om. Consequently, the Trudinger--Moser functional has no nonnegative constrained critical points at sufficiently large prescribed energies. The proof combines a weighted Pohozaev identity obtained after conformal change of variables, concentration analysis of normalized measures, and local radial comparison. The comparison hypotheses are obtained from local integral estimates, without an a priori energy bound. An estimate for the exterior Green potential then excludes sequences of unbounded energy.
ER -