Trudinger-Moser Embedding on the Hyperbolic Space

Yunyan Yang; Xiaobao Zhu

doi:10.1155/2014/908216

2014 Trudinger-Moser Embedding on the Hyperbolic Space

Yunyan Yang, Xiaobao Zhu

Abstr. Appl. Anal. 2014: 1-4 (2014). DOI: 10.1155/2014/908216

Abstract

Let $({\Bbb H}^{n},g)$ be the hyperbolic space of dimension $n$ . By our previous work (Theorem 2.3 of (Yang (2012))), for any $\mathrm{0}<\alpha <{\alpha }_{n}$ , there exists a constant $\tau >\mathrm{0}$ depending only on $n$ and $\alpha$ such that ${\text{s}\text{u}\text{p}}_{u\in {W}^{1,n}({\Bbb H}^{n}),{\parallel u\parallel }_{1,\tau }\le 1}{\int }_{\Bbb H}^{n}({e}^{{\alpha |u|}^{n/(n-1)}}-{\sum }_{k=0}^{n-2}{\alpha }^{k}{|u|}^{nk/(n-1)}/k\text{!}){dv}_{g}<\infty ,$ where ${\alpha }_{n}=n{\omega }_{n-\mathrm{1}}^{\mathrm{1}/(n-\mathrm{1})}$ , ${\omega }_{n-\mathrm{1}}$ is the measure of the unit sphere in ${\Bbb R}^{n}$ , and ${ {\parallel} u{\parallel}}_{1,\tau }={{\parallel}{\nabla }_{g}u{\parallel}}_{{L}^{n}({\Bbb H}^{n})}+\tau {{\parallel}u{\parallel}}_{{L}^{n}({\Bbb H}^{n})}$ . In this note we shall improve the above mentioned inequality. Particularly, we show that, for any $\mathrm{0}<\alpha <{\alpha }_{n}$ and any $\tau >\mathrm{0}$ , the above mentioned inequality holds with the definition of ${{\parallel}u{\parallel}}_{1,\tau }$ replaced by $({\int }_{{\Bbb H}^{n}}\mathrm{‍}(|{\nabla }_{g}u{|}^{n}+\tau |u{|}^{n})d{v}_{g}{)}^{\mathrm{1}/n}$ . We solve this problem by gluing local uniform estimates.

Citation

Download Citation

Yunyan Yang. Xiaobao Zhu. "Trudinger-Moser Embedding on the Hyperbolic Space." Abstr. Appl. Anal. 2014 1 - 4, 2014. https://doi.org/10.1155/2014/908216