Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in R^(2)  

作　　者：Wei Wei SHAN Xiao Meng LI 

机构地区：[1]School of Mathematical Sciences,Huaibei Normal University,Huaibei,235000,P.R.China [2]School of Mathematics and Big Data,Chaohu University,Hefei,230000,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2024年第11期2840-2854,共15页数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(Grant No.12201234);the Natural Science Foundation of Anhui Province of China(Grant No.2008085MA07)。

摘　　要：Let B be a unit ball in R^(2),W_(0)^(1,2)(B)be the standard Sobolev space.For anyϵ>0,de Figueiredo,doÓ,dos Santons,Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball.Precisely,u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)^(|x|^(2∈)e^(4π(1+∈)u^(2)dx)can be attained by some radially symmetric function u_(∈)∈W_(0)^(1,2)(B)∫_(B)with∫_(B)|▽u|^(2)dx=1.In this note,we concern the compactness of the function family{uϵ}ϵ>0 and prove that up to a subsequence uϵconverges to some function u 0 in C^(1)(B)asϵ→0.Furthermore,u_(0) is an extremal function of the supremum u∈W_(0)^(1,2)(B)^(sup),∫_(B)|▽u|^(2)dx≤1∫_(B)e^(4πu^(2)dx)Let us explain the result in geometry.Denoteω_(0)=dx_(1)^(2)+dx_(2)^(2)be the standard Euclidean metricω_(∈)|x|^(2ϵ)ω_(0)for xϵB..Then the extremal family{u_(∈)}∈>0 of the following Trudinger-Moser functionals∫_(B)e^(4π(1+∈)u^(2))dvω_(∈)under the constraint W_(0)^(1,2)(B)and∫_(B)|▽ω_∈)u|^(2)dvω_∈)≤1 is compact asϵ→0.

关 键 词：Trudinger-Moser inequality extremal function blow-up analysis COMPACTNESS 

分 类 号：O177[理学—数学]