On the bounds of sharp Trudinger-Moser inequalities

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.