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

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（12）
• 页码范围: 2497-2504
• 页数: 8页
• 语种: 英文

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∈H10,rad(B)‖▽u‖L2(B)=1 } is attained,where B denotes the unit ball in RN(N ≥ 3),μ ∈ (0,N),p(r) =2*μ + rt,t ∈ (0,min{N/2-μ/4,N-2}) and 2*μ =(2N-μ)/(N-2) is the critical exponent for the Hardy-Littlewood-Sobolev inequality.