A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains
Year: 2026
Type: preprint
Abstract: \noindent Let \(\Om\subset\R^2\) 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.
Source: arXiv (Cornell University)
Authors: Hu, Hongtao, Wu, Qiaoran, Zou, Wenming
Institution: Tsinghua University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3156265683
OpenAlex ID: W7221043663