Anisotropic Moser-Trudinger Inequality Involving L^(n) Norm in the Entire Space R^(n)  

作　　者：Ru Long XIE 

机构地区：[1]Department of Mathematics,Chaohu University,Hefei 238000,P.R.China [2]School of Mathematical Sciences,University of Science and Technology of China,Hefei 236000,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2023年第12期2427-2451,共25页数学学报（英文版）

基　　金：Supported by Natural Science Foundation of China(Grant Nos.11526212,11721101,11971026);Natural Science Foundation of Anhui Province(Grant No.1608085QA12);Natural Science Foundation of Education Committee of Anhui Province(Grant Nos.KJ2016A506,KJ2017A454);Excellent Young Talents Foundation of Anhui Province(Grant No.GXYQ2020049)。

摘　　要：Let F:R^(n)-→[0,+∞)be a convex function of class C^(2)(R^(n)/{0})which is even and positively homogeneous of degree 1,and its polar F0 represents a Finsler metric on R^(n).The anisotropic Sobolev norm in W^(1,n)(R^(n))is defined by||u||F=(∫_(R_(n)(F^(n)(↓△u)+|u|^(n)dx)^(1/n)In this paper,the following sharp anisotropic Moser-Trudinger inequality involving L^(n)norm u∈W^(1,n)^(SUP)(R^(n),||u||F≤1∫_(R^(n))Ф(λ_(n)|u|n/n-1(1+a||u||^(n)_(n)1/n-1)dx<+∞in the entire space R^(n)for any 0<a<1 is estabished,whereФ(t)=e^(t)-∑^(n-2)_(j=0)tj/j!,λ_(n)=n^(n/n-1)k_(n)1/n-1 and kn is the volume of the unit Wulf ball in Rn.It is also shown that the above supremum is infinity for all α≥1.Moreover,we prove the supremum is attained,that is,there exists a maximizer for the above supremum whenα>O is sufficiently small.

关 键 词：Moser-Trudinger inequality anisotropic Sobolev norm blow up analysis extremal func-tion unbounded domain 

分 类 号：O178[理学—数学]