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

\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