Achievability of a supremum for the Hardy-Littlewood-Sobolev inequality with supercritical exponent  

作　　者：Xiaoming An Shuangjie Peng Chaodong Xie 

机构地区：[1]School of Mathematics and Statistics,Hubei Key Laboratory of Mathematical Sciences,Central China Normal University,Wuhan 430079,China [2]School of Economics Management,Guizhou University for Ethinic Minorities,Guiyang 550025,China

出　　处：《Science China Mathematics》2019年第12期2497-2504,共8页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11831009 and 11571130)

摘　　要：In this paper, we prove that the supremum sup{ ∫B∫B|u（y）|p（|y|）|u（x）|p（|x|）/|x-y|μdxdy : u ∈ H0,rad1（B）, ||?||uL2（B）= 1}is attained, where B denotes the unit ball in RN（N ≥3）, μ ∈（0, N）, p（r） = 2μ*+ rt, t ∈（0, min{N/2-μ/4, N-2}） and 2μ*=（2N-μ）/（N-2） is the critical exponent for the Hardy-Littlewood-Sobolev inequality.In this paper,we prove that the supremum sup{∫B∫B|u(y)|^p(|y|)|u(x)|^p(|x|)/|x-y|^u dxdy:u∈H0^1,rad(B),‖▽u‖L^2(B)=1}is attained,where B denotes the unit ball in R^N(N≥3),μ∈(0,N),p(r)=2^xμ+r^t,t∈(0,min{N/2-μ/4,N-2})and 2μ^*=(2N-μ)/(N-2)is the critical exponent for the Hardy-Littlewood-Sobolev inequality.

关 键 词：Hardy-Littlewood-Sobolev INEQUALITY achievability of a SUPREMUM SUPERCRITICAL EXPONENT 

分 类 号：O17[理学—数学]