A priori Dirichlet energy bounds for positive Trudinger--Moser critical points on simply connected planar domains

\noindent Let $\text{Om}\subset {\mathbb{R}}^2$ be a smooth bounded simply connected domain. We prove that all positive solutions of $-\mathrm{\Delta }u=\theta u{e}^{u^2}$, with $\theta >0$ and zero Dirichlet boundary data, have Dirichlet energy bounded by a constant depending only on $\text{Om}$. 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.