Threshold Attainment and Sharp Asymptotics for Lp-Perturbed Moser--Trudinger Functionals on ℝ2
Abstract
In this paper, we study the critical Moser--Trudinger supremum S(λ,p) = supsubstack u∈ H1(ℝ2) ∫ℝ2 (|∇ u|2+u2) dx≤ 1 ∫ℝ2 (e4πu2−1−λ|u|p) dx, p≥ 2. Building on the classification away from the thresholds by Chen, Jiang, Lu and Zhu (2026), we resolve the endpoint cases: both thresholds for p=2 and the finite threshold for 2<p≤4 are attained. We also prove compactness of centred radial extremals and one-sided value expansions governed by the selected endpoint Lp mass. For p>4, we quantify how extremals spread out as λ→∞ by establishing the two-term expansion S(λ,p)=4π+𝒞pλ−2/(p−4) +Dpλ−4/(p−4)+o(λ−4/(p−4)), Here 𝒞p is the maximum of a limiting variational problem at unit L2 mass. An explicit correction involving the sixth moment and the full H1 normalization determines Dp and selects the limiting maximizers. Rescaled centred radial extremals converge strongly to these profiles in H1(ℝ2) along subsequences. For every fixed λ>λ*(4), we further establish limp↓4(p−4)log(S(λ,p)−4π) =2logλloc/λ<0, where λloc is the local critical penalty determined by the sharp Gagliardo--Nirenberg constant. The gradient energy has the same logarithmic rate, and the centred radial profiles, normalized in mass and gradient energy, converge along subsequences to sharp Gagliardo--Nirenberg optimizers. An exact transfer identity extends these endpoint and asymptotic results to every constant planar norm.
Related
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BibTeX
@misc{guo2026threshold,
title = {{Threshold Attainment and Sharp Asymptotics for $L^p$-Perturbed Moser--Trudinger Functionals on $\mathbbR^2$}},
author = {Kaiwen Guo and Yanjun Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29402},
eprint = {2610.29402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29402},
}
RIS
TY - UNPB
TI - Threshold Attainment and Sharp Asymptotics for L^p-Perturbed Moser--Trudinger Functionals on ℝ^2
AU - Kaiwen Guo
AU - Yanjun Liu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29402
UR - https://arxiv.org/abs/2610.29402
AB - In this paper, we study the critical Moser--Trudinger supremum S(λ,p) = sup_substack u∈ H^1(ℝ^2) ∫_ℝ^2 (|∇ u|^2+u^2) dx≤ 1 ∫_ℝ^2 (e^4πu^2−1−λ|u|^p) dx, p≥ 2. Building on the classification away from the thresholds by Chen, Jiang, Lu and Zhu (2026), we resolve the endpoint cases: both thresholds for p=2 and the finite threshold for 2<p≤4 are attained. We also prove compactness of centred radial extremals and one-sided value expansions governed by the selected endpoint L^p mass. For p>4, we quantify how extremals spread out as λ→∞ by establishing the two-term expansion S(λ,p)=4π+𝒞_pλ^−2/(p−4) +D_pλ^−4/(p−4)+o(λ^−4/(p−4)), Here 𝒞_p is the maximum of a limiting variational problem at unit L^2 mass. An explicit correction involving the sixth moment and the full H^1 normalization determines D_p and selects the limiting maximizers. Rescaled centred radial extremals converge strongly to these profiles in H^1(ℝ^2) along subsequences. For every fixed λ>λ_*(4), we further establish lim_p↓4(p−4)log(S(λ,p)−4π) =2logλ_loc/λ<0, where λ_loc is the local critical penalty determined by the sharp Gagliardo--Nirenberg constant. The gradient energy has the same logarithmic rate, and the centred radial profiles, normalized in mass and gradient energy, converge along subsequences to sharp Gagliardo--Nirenberg optimizers. An exact transfer identity extends these endpoint and asymptotic results to every constant planar norm.
ER -