A Remark on Hardy-Trudinger-Moser Inequality

• 来源期刊: 偏微分方程（英文版）
• 学科: 数学
• 关键词: Hardy-Trudinger-Moser inequality Trudinger-Moser inequality blow-up analysis
• 年，卷（期）: 2018,（4）
• 页码范围: 353-373
• 页数: 21页
• 分类号: O177
• 语种: 英文
• DOI: 10.4208/jpde.v31.n4.6

摘要

Let B be the unit disc in R2,(H) be the completion of C∞0(B) under the norm ‖u‖(H)=(∫B|▽u|2dx-∫Bu2/(1-|x|2)2dx)1/2,(∨)u∈C∞0(B).Using blow-up analysis,we prove that for any γ ≤ 4π,the supremum supu∈(H),‖u‖1,h≤1∫Beγu2dx can be attained by some function u0 ∈ (H) with ‖u0‖1,h =1,where h is a decreasingly nonnegative,radially symmetric function,and satisfies a coercive condition.Namely there exists a constant δ ＞ 0 satisfying ‖u‖21,h=‖u‖2(H)-∫Bhu2dx≥δ‖u‖2(H),(∨)u∈(H).This extends earlier results of Wang-Ye [1] and Yang-Zhu [2].