Existence of extremals for Moser-Trudinger inequalities with a decaying potential in Hyperbolic space Existence of extremals for Moser-Trudinger

• Published in: Manuscripta mathematica Vol. 176; no. 4
• Main Authors: Sun, Jingxuan, Song, Zhen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 14.08.2025
• Subjects: Algebraic Geometry; Calculus of Variations and Optimal Control; Optimization; Geometry; Lie Groups; Mathematics; Mathematics and Statistics; Number Theory; Topological Groups; 35J20; 46E35; 35J60
• ISSN: 0025-2611; 1432-1785
• DOI: 10.1007/s00229-025-01660-9

This paper mainly shows the existence or non-existence of extremals for the Moser-Trudinger inequality in Hyperbolic space. We demonstrate that the following classical Moser-Trudinger inequality on Hyperbolic space S ( α ) : = sup ‖ ∇ g u ‖ 2 ≤ 1 ∫ H 2 e α u 2 - 1 d v g < ∞ holds if and only if α ∈ ( 0 , 4 π ] . For α ∈ ( 0 , α ∗ ) , we prove that S ( α ) = 4 α , and it can not be attained by any extremal functions, where α ∗ is a positive constant given in Lemma 3.1 . Besides, we consider the Moser-Trudinger inequality with a decaying potential. We prove that for any α ∈ ( 0 , 4 π ] , there exists v ∈ W 1 , 2 ( H 2 ) with ‖ v ‖ V = 1 such that S ( V , α ) : = sup ‖ u ‖ V ≤ 1 ∫ H 2 e α u 2 - 1 d v g = ∫ H 2 e α v 2 - 1 d v g < ∞ . Here, 0.1 ‖ u ‖ V 2 : = ∫ H 2 | ∇ g u | g 2 - V ( x ) | u | 2 d v g , and V : H 2 → R is a decaying potential satisfying the following condition ( V 1 ) V ( x ) = 1 - | x | 2 4 + V ~ ( x ) , where 0 = inf x ∈ H 2 V ~ ( x ) = lim | x | → 1 V ~ ( x ) < sup x ∈ H 2 V ~ ( x ) = l < 1 4 . Our result is a partial answer to the open question in [ 32 ], and is an analog of the celebrated result of Carleson–Chang [ 9 ] for the Moser–Trudinger inequality and the result of Wang and Ye [ 52 ] for Hardy–Moser–Trudinger inequality.