The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints

• Published in: Mathematische annalen 2021-08, Vol.380 (3-4), p.1933-1958
• Main Author: Nguyen, Van Hoang
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Thresholds
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-020-02010-8

This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case M T N ( a , β ) = sup u ∈ W 1 , N ( R N ) , ‖ ∇ u ‖ N a + ‖ u ‖ N N = 1 ∫ R N Φ N ( 1 - β / N ) α N | u | N N - 1 | x | - β d x , where a > 0 , β ∈ [ 0 , N ) , Φ N ( t ) = e t - ∑ k = 0 N - 2 t k k ! , α N = N ω N - 1 1 / ( N - 1 ) , and ω N - 1 denotes the surface area of the unit sphere in R N . More precisely, we study the effect of the parameter a to the attainability of M T N ( a , β ) . We will prove that for each β ∈ [ 0 , N ) there exist the thresholds a ∗ ( β ) and a ∗ ( β ) such that M T N ( a , β ) is attained for any a ∈ ( a ∗ ( β ) , a ∗ ( β ) ) and is not attained for a < a ∗ ( β ) or a > a ∗ ( β ) . We also give some qualitative estimates for a ∗ ( β ) and a ∗ ( β ) . Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do Ó, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).