Extremal Functions for Adams Inequalities in Dimension Four

• 作者: LI Xiaomeng
• 来源期刊: 偏微分方程（英文版）
• 学科: 数学
• 关键词: Adams inequality Trudinger-Moser inequality extremal function blow-up analysis
• 年，卷（期）: 2020,（2）
• 页码范围: 171-192
• 页数: 22页
• 分类号: O17
• 语种: 英文
• DOI: 10.4208/jpde.v33.n2.6

摘要

Let Ω (∈) R4 be a smooth bounded domain,W2,20(Ω) be the usual Sobolev space.For any positive integer e,λl(Ω) is the e-th eigenvalue of the bi-Laplacian operator.Define E(l)=Eλ1(Ω) (⊕ Eλ2(Ω)(⊕) … ⊕Eλe(Ω),where Eλi(Ω) is eigenfunction space associated with λi(Ω).E⊥e denotes the orthogonal complement of Ee in W2,20 (Ω).For 0 ≤ α ≤ λe+1 (Ω),we define a norm by ||u||22,α =||△u||22-α||u||22 for u ∈ E⊥e.In this paper,using the blow-up analysis,we prove the following Adams inequalities supu∈E⊥e,||u||2,α≤1∫Ω e32π2u2 dx ＜ +∞;moreover,the above supremum can be attained by a function u0 ∈ E⊥e∩C4((Ω)) with || u0||2,α =1.This result extends that of Yang (J.Differential Equations,2015),and complements that of Lu and Yang (Adv.Math.2009) and Nguyen (arXiv: 1701.08249,2017).