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The compactness of Moser-Trudinger type inequalities in the unit ball
Abstract
In this paper, we employ the concentration-compactness principle to show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫𝔹 |x|N ε Φ( αN ( 1+ ε) |u| N/(N−1) ) dx over all such functions is uniformly bounded. Furthermore, we also prove the existence of extremals. Finally, we consider the compactness of the sequence of extremals of the inequalities and the limit of this sequence is the extremal of ∫𝔹 Φ( αN |u| N/(N−1) ) dx in C1 ( 𝔹 ), where $α_N = N ω_{N-1}^{ \frac{1}{N-1} } $, $Φ\left( t \right) := e^t - \sum_{j=0}^{N-2} \frac{t^j}{j!} $ and ωN−1 is the surface of the unit ball in ℝN.
Related
- Semiclassical Moser--Trudinger type inequalities for orthonormal families in ℝ2
- On sharp singular Moser-Trudinger type inequalities and applications to zero mass (p,Q)-Laplace subelliptic equations in the Heisenberg group
- Extremals for Trudinger-Moser inequalities via polygonal profiles
- Best Constants for Anisotropic Moser-Trudinger Inequalities with the Exact Growth Condition and Their Applications
- Sharp singular Moser-Trudinger inequalities and n-Laplacian equations with degenerate potentials
BibTeX
@misc{xia2026compactness,
title = {{The compactness of Moser-Trudinger type inequalities in the unit ball}},
author = {Qi Xia and Yufeng Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37363},
eprint = {2609.37363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37363},
}
RIS
TY - UNPB
TI - The compactness of Moser-Trudinger type inequalities in the unit ball
AU - Qi Xia
AU - Yufeng Lu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37363
UR - https://arxiv.org/abs/2609.37363
AB - In this paper, we employ the concentration-compactness principle to show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫_𝔹 |x|^N ε Φ( α_N ( 1+ ε) |u|^ N/(N−1) ) dx over all such functions is uniformly bounded. Furthermore, we also prove the existence of extremals. Finally, we consider the compactness of the sequence of extremals of the inequalities and the limit of this sequence is the extremal of ∫_𝔹 Φ( α_N |u|^ N/(N−1) ) dx in C^1 ( 𝔹 ), where $α_N = N ω_{N-1}^{ \frac{1}{N-1} } $, $Φ\left( t \right) := e^t - \sum_{j=0}^{N-2} \frac{t^j}{j!} $ and ω_N−1 is the surface of the unit ball in ℝ^N.
ER -