# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.02349