Best constants and existence of maximizers for weighted Trudinger–Moser inequalities

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• 作者：Mengxia Dong ; Guozhen Lu
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：55
• 期：4
• 全文大小：593 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835
• 卷排序：55

文摘

The main purpose of this paper is three-fold. First of all, we will establish a weighted Trudinger–Moser inequality on the entire space without requiring the functions under consideration being radially symmetric (see Theorem 1.1). We will also prove the existence of a maximizer of this sharp weighted inequality. Therefore, our result improves the earlier one where such type of inequality has only been proved for spherically symmetric functions by M. Ishiwata, M. Nakamura, H. Wadade in (Ann Inst H Poincaré Anal Non Linaire 31(2):297–314, 2014) (except in the case \(s\not =0\)). Second, we will prove stronger weighted Trudinger–Moser inequalities for functions which are not required to be radially symmetric (see Theorems 1.2 and 1.3). Finally, the existence of a maximizer of these stronger inequalities is proved in Theorem 1.4.Mathematics Subject Classification46E3042B3546E3542B37