Uniform parameter bounds for weighted exponential equations on simply connected domains

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 $1<p<2$ without a priori energy bound.