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 \(\begin{aligned} S(\alpha ):=\sup _{\Vert \nabla _g u\Vert _2\le 1}\int _{{\mathbb {H}}^2}e^{\alpha u^2}-1 \,dv_g <\infty \end{aligned}\) holds if and only if \(\alpha \in (0,4\pi ]\) . For \(\alpha \in (0,\alpha ^*)\) , we prove that \(S(\alpha )=4\alpha \) , and it can not be attained by any extremal functions, where \(\alpha ^*\) 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 \(\alpha \in (0,4\pi ]\) , there exists \(v\in W^{1,2}({\mathbb {H}}^2)\) with \(\Vert v\Vert _V=1\) such that \(\begin{aligned} S(V,\alpha ):=\sup _{\Vert u\Vert _V\le 1}\int _{{\mathbb {H}}^2}e^{\alpha u^2}-1 \,dv_g=\int _{{\mathbb {H}}^2}e^{\alpha v^2}-1 \,dv_g<\infty . \end{aligned}\) Here, 0.1 \(\begin{aligned} \Vert u\Vert ^2_V:=\int _{{\mathbb {H}}^2}|\nabla _g u|_g^2-V(x)|u|^2 \,dv_g, \end{aligned}\) and \(V:{\mathbb {H}}^2\rightarrow {\mathbb {R}}\) is a decaying potential satisfying the following condition \((V_1)\) \(V(x)=\frac{1-|x|^2}{4}+{\widetilde{V}}(x)\) , where \(\begin{aligned} 0=\inf _{x\in {\mathbb {H}}^2} {\widetilde{V}}(x)=\lim _{|x|\rightarrow 1}{\widetilde{V}}(x)<\sup _{x\in {\mathbb {H}}^2} {\widetilde{V}}(x)=l<\frac{1}{4}. \end{aligned}\) 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.
