Uniform parameter bounds for weighted exponential equations on simply connected domains

We consider positive solutions of {equation*} {cases} -Δu=pλh(x)uᵖ⁻¹eu^p&in Ω,\\ u>0&in Ω,\\ u=0&on ∂Ω, {cases} {equation*} where p∈[1,2], $λ>0$, Ω² is a smooth bounded simply connected domain and h∈ C²(Ω) is positive on Ω. The cases $p=1$ and $p=2$ correspond to the Liouville and Moser-Trudinger equations, respectively. We prove that the normalized parameter {equation*} β=λp^2/2(∫_Ωh(eu^p-1)dx)2-p/p(∫_Ωhu^peu^pdx)2(p-1)/p {equation*} is bounded by a constant depending only on $Ω$ and h, uniformly over all p∈[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≡1, we further prove that the energy converges uniformly to $4π$ as λ→0 when $Ω$ is convex. We also develop a boundary exclusion argument for positive weights that rules out boundary blow-up for $1