Uniform parameter bounds for weighted exponential equations on simply connected domains
Abstract
We consider positive solutions of −Δu=pλh(x)up−1eup&in Ω, u>0&in Ω, u=0&on ∂Ω, where p∈[1,2], λ>0, Ω⊂ℝ2 is a smooth bounded simply connected domain and h∈ C2(Ω̅) is positive on Ω̅. The cases p=1 and p=2 correspond to the Liouville and Moser-Trudinger equations, respectively. We prove that the normalized parameter β=λp2/2(∫Ωh(eup−1)dx)(2−p)/p(∫Ωhupeupdx)(2(p−1))/p 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<p<2 without a priori energy bound.
Related
- Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter
- A weighted semigroup approach to exponential stability in linear parabolic equations
- Discontinuous Galerkin Approximation to a Class of Strongly Damped Wave Equations with Uniform Exponential Decay
- Uniform Exponential Stability for Finite Energy Weak Solutions to the Barotropic Compressible Navier--Stokes Equations with Large Potential External Forces
- Uniform Observability of High-Order Fully Discrete Multidimensional Wave Equations: Exponential Loss and Recovery
BibTeX
@misc{chen2026uniform,
title = {{Uniform parameter bounds for weighted exponential equations on simply connected domains}},
author = {Hong-Ge Chen and Peng Luo and Shuangjie Peng and Fulin Zhong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37177},
eprint = {2610.37177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37177},
}
RIS
TY - UNPB
TI - Uniform parameter bounds for weighted exponential equations on simply connected domains
AU - Hong-Ge Chen
AU - Peng Luo
AU - Shuangjie Peng
AU - Fulin Zhong
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37177
UR - https://arxiv.org/abs/2610.37177
AB - We consider positive solutions of −Δu=pλh(x)u^p−1e^u^p&in Ω, u>0&in Ω, u=0&on ∂Ω, where p∈[1,2], λ>0, Ω⊂ℝ^2 is a smooth bounded simply connected domain and h∈ C^2(Ω̅) is positive on Ω̅. The cases p=1 and p=2 correspond to the Liouville and Moser-Trudinger equations, respectively. We prove that the normalized parameter β=λp^2/2(∫_Ωh(e^u^p−1)dx)^(2−p)/p(∫_Ωhu^pe^u^pdx)^(2(p−1))/p 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<p<2 without a priori energy bound.
ER -