Singular Moser-Trudinger inequality with the exact growth condition on hyperbolic space

Zhao Liu,Lu Chen

doi:10.1186/s13660-017-1398-8

Abstract

In this paper, we are concerned with a singular version of the Moser-Trudinger inequality with the exact growth condition in the n-dimension hyperbolic space mathbb{H}^{n}. Our result is a natural extension of the work of Lu and Tang in (J. Geom. Anal. 26:837-857, 2016).

Highlights

Let W,p( ) denote the usual Sobolev space, i.e., the completion of C ∞( ) under the Sobolev norm u W,p( ) = p|∇u|p + |u|p dx .The classical Sobolev embedding theorem states W,n( ) ⊂ Lq( ) for all ≤ q < ∞ but W,n( ) L∞( ).One can check this by choosing the function u(x) = ln(ln R |x–x
This paper is concerned with a singular version of Moser-Trudinger inequality with the exact growth condition on hyperbolic space Hn
Motivated by the above results, we consider the singular version of the Moser-Trudinger inequality with the exact growth condition on hyperbolic space

Summary

Introduction

A natural idea is to consider the Moser-Trudinger inequality in the whole space Rn. Adachi and Tanaka in [ ] proved the following nice result. |∇u|n + |u|n dx , Rn and obtained the Moser-Trudinger inequality in the whole space Rn in the case of α = αn. This paper is concerned with a singular version of Moser-Trudinger inequality with the exact growth condition on hyperbolic space Hn. The hyperbolic space Hn (n ≥ ) is a complete and connected Riemannian manifold and has constant curvature equal to – .

Results

Conclusion