Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Abstract

Let $\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}$ and $\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}$ be the sharp constant in Moser’s inequality (where $\omega _{n-1}$ is the area of the surface of the unit $n$-ball in $\mathbb {R}^n$), and $dV$ be the volume element on the $n$-dimensional hyperbolic space $(\mathbb {H}^n, g)$ ($n\ge {2}$). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on $\mathbb {H}^n$:

For any $u\in {W^{1,n}(\mathbb {H}^n)}$ satisfying $\Vert \nabla _{g}u\Vert _{n}\le {1}$, there exists a constant $C(n)>0$ such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power $\frac{n}{n-1}$ and the constant $\alpha _{n}$ are optimal in the following senses:

• (i) If the power $\frac{n}{n-1}$ in the denominator is replaced by any $p<\frac{n}{n-1}$, then there exists a sequence of functions $\{u_{k}\}$ such that $\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}$, but

  $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$

• (ii) If $\alpha >\alpha _{n}$, then there exists a sequence of function $\{u_{k}\}$ such that $\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}$, but

  $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

  for any $p\ge {0}$.

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.