{ "abstract": "We consider positive solutions of \\begin{equation*} \\begin{cases} -Δu=pλh(x)u^{p-1}\\mathrm{e}^{u^p}&\\text{in }Ω,\\\\ u>0&\\text{in }Ω,\\\\ u=0&\\text{on }\\partialΩ, \\end{cases} \\end{equation*} where $p\\in[1,2]$, $λ>0$, $Ω\\subset\\mathbb{R}^2$ is a smooth bounded simply connected domain and $h\\in C^2(\\overlineΩ)$ is positive on $\\overlineΩ$. The cases $p=1$ and $p=2$ correspond to the Liouville and Moser-Trudinger equations, respectively. We prove that the normalized parameter \\begin{equation*} β=\\frac{λp^2}{2}\\left(\\int_Ωh(\\mathrm{e}^{u^p}-1)\\mathrm{d}x\\right)^{\\frac{2-p}{p}}\\left(\\int_Ωhu^p\\mathrm{e}^{u^p}\\mathrm{d}x\\right)^{\\frac{2(p-1)}{p}} \\end{equation*} is bounded by a constant depending only on $Ω$ and $h$, uniformly over all $p\\in[1,2]$, $λ>0$ and positive solutions. At $p=2$, this excludes positive constrained critical points of the weighted Moser-Trudinger functional at sufficiently large Dirichlet energies. For $p=2$ and $h\\equiv1$, we further prove that the energy converges uniformly to $4π$ as $λ\\to0$ when $Ω$ is convex. We also develop a boundary exclusion argument for positive weights that rules out boundary blow-up for $1