On the bounds of sharp Trudinger-Moser inequalities

TL;DR

This paper establishes bounds for sharp Trudinger-Moser inequalities on Euclidean space, providing new upper and lower bounds, analyzing singular cases, and exploring asymptotic behaviors to advance understanding of these inequalities.

Contribution

It introduces new bounds for sharp Trudinger-Moser inequalities, including singular cases and asymptotic analysis, improving previous results by Lam Lu and Zhang.

Findings

Bounds for TM(B) are between 2.15(n-1) and 36n-35.

For large n, bounds are between 2.15(n-1) and 11.5n-10.5.

Upper bounds for subcritical and critical inequalities are provided.

Abstract

In this paper, we establish the bounds of sharp Trudinger-Moser inequalities on Euclidean space. Let $B$ be a ball in $\mathbb{R}^n$ and $$TM(B)=\sup_{u\in{W_{0}^{1,n}(B)},\|\nabla u\|_{n}\leq{1}}\frac{1}{|B|}\int_{B}\exp(\alpha_{n}|u(x)|^{\frac{n}{n-1}})dx.$$ We prove that $$2.15(n-1)\leq TM(B)\leq{36n-35}.$$ If $n$ is large enough, we have $$2.15(n-1)\leq TM(B)\leq{11.5n-10.5}.$$ Singular case are also considered. Moreover we provide the upper bounds for subcritical and critical Trudinger-Moser inequalities respectively. At last we study the asymptotically behavior of subcritical Trudinger-Moser inequalities, which improve Lam Lu and Zhang's work.