Energy Estimate Related to a 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 energy estimate blow-up analysis
• 年，卷（期）: 2019,（4）
• 页码范围: 342-351
• 页数: 10页
• 分类号: O17
• 语种: 英文
• DOI: 10.4208/jpde.v32.n4.4

摘要

Let B1 be a unit disc of R2,and H be a completion of C0∞(B1) under the norm ||u||H2 =∫B1 (|▽u|2-u2/(1-|x|2)2) dx.Using blow-up analysis,Wang-Ye [1] proved existence of extremals for a Hardy-Trudinger-Moser inequality.In particular,the supremumu∈H,sup||u||H≤∫B1e4πru2dx can be attained by some function u0 ∈ H with ||u0||H =1.This was improved by the author and Zhu [2] to a version involving the first eigenvalue of the Hardy-Laplacian operator-△-1/(1-|x|2)2.In this note,the results of [1,2] will be reproved by the method of energy estimate,which was recently developed by Malchiodi-Martinazzi [3] and Mancini-Martinazzi [4].