发表机构
Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明单连通平面域上正 Trudinger--Moser 临界点的 Dirichlet 能量有先验界，从而排除了大能量下的非负临界点，方法结合 Pohozaev 恒等式、集中分析和 Green 势估计。
AI 中文摘要
设 \\(\Om\subset\R^2\\) 为光滑有界单连通域。我们证明所有满足 \\(-\Delta u=\theta ue^{u^2}\\)（其中 \\(\theta>0\\)，且具有零 Dirichlet 边界数据）的正解，其 Dirichlet 能量被一个仅依赖于 \\(\Om\\) 的常数所界定。因此，Trudinger--Moser 泛函在足够大的给定能量下不存在非负约束临界点。证明结合了共形变量替换后得到的加权 Pohozaev 恒等式、归一化测度的集中分析以及局部径向比较。比较假设由局部积分估计获得，无需先验能量界。随后，外部 Green 势的估计排除了无界能量的序列。
英文摘要
\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.
