Year: 2026
Type: preprint
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 $$ \int_{\mathbb{B}} |x|^{N ε} Φ\left( α_N \left( 1+ ε\right) |u|^{ \frac{N}{N-1} } \right) dx $$ 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 $$ \int_{\mathbb{B}} Φ\left( α_N |u|^{ \frac{N}{N-1} } \right) dx $$ in $C^1 \left( \mathbb{B} \right)$, 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 $\mathbb{R}^N$.
Source: arXiv (Cornell University)
Authors: Xia, Qi, Lu, Yufeng
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8232460072
OpenAlex ID: W7214889629