This paper is concerned with the generalization of the Moser–Trudinger inequality in the hyperbolic space. We establish an improved Moser–Trudinger inequality in the hyperbolic space, i.e., for any \(n\ge 2,0\le \beta < n,V\in {\cal{P}}\) and u ∈ W1,n(ℍn), we have \(\mathop {\sup}\limits_{{{\| u \|}_{\cal{H}}} \le 1} \int_{{\mathbb{H}^n}} {{{{\Phi _{n - 1}}({{\alpha _n}({1 - {\beta \over n}}){{| u |}^{{n \over {n - 1}}}}})} \over {{{[{d({0,x})}]}^\beta}}}} dV < \infty,\) where \(\alpha_{n}=n\omega_{n-1}^{{1\over{n-1}}},\omega_{n-1}\) denotes the surface area of the unit sphere in ℝn, \(d(x,0)=\ln{1+\vert x\vert\over{1-\vert x\vert}}\) denotes the geodesic distance from x to 0 and \({\Phi _n}(t) = {e^t} - \sum\limits_{k = 0}^{n - 1} {{{{t^k}} \over {k!}}}.\) Besides, the norm \(\Vert\cdot\Vert_{\cal{H}}\) and the set of the potential functions \({\cal{P}}\) are defined by \(\| u \|_{\cal{H}}^n: = \int_{{\mathbb{H}^n}} {{{| {{\nabla_\mathbb{H}}u} |}^n}} dV + \int_{{\mathbb{H}^n}} {V(x)} {| u |^n}dV\) and \({\cal{P}} = \left\{{V:{\mathbb{H}^n} \to \left[{- {{\left({{{n - 1} \over n}} \right)}^n}, + \infty} \right):\exists \delta > 0\,{\rm{s.t.}\,{\text{Vol}}_g}\left\{{V < - {{\left({{{n - 1} \over n}} \right)}^n} + \delta} \right\} < \infty} \right\},\) where Volg{·} denotes the volume with respect to the hyperbolic metric. Moreover, with the help of the Hardy inequality and Poincaré–Sobolev inequality in the hyperbolic space, we establish Moser–Trudinger inequalities with weaker constraints: the potentials V: ℍn → ℝ need not be greater than \(-({n-1\over{n}})^{n}\) and even can tend to −∞ (see Theorem 1.3 and Theorem 1.5). This phenomenon violates the \(-({n-1\over{n}})^{n}\) lower bound constraint for the trapping potential, and to the best of our knowledge, this paper is the first to establish the Moser–Trudinger inequality for potentials which may tend to −∞ in n-dimensional hyperbolic space.
