On the bounds of sharp Trudinger-Moser inequalities

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 $$TM(B)=\underset{u\in W_0^{1,n}(B),\Vert \nabla u{\Vert }_n\le 1}{\sup }\frac{1}{|B|}{\int }_B\exp ({\alpha }_n|u(x){|}^{\frac{n}{n-1}})dx.$$ We prove that $$2.15(n-1)\le TM(B)\le 36n-35.$$ If $n$ is large enough, we have $$2.15(n-1)\le TM(B)\le 11.5n-10.5.$$ 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.

arXiv commentthere is an error in section 3