A sharp Trudinger-Moser type inequality involving $L^{n}$ norm in the entire space $mathbb{R}^{n}$

Let $W^{1,n} ( mathbb{R}^{n} $ be the standard Sobolev space and $leftVert cdotrightVert _{n}$ be the $L^{n}$ norm on $mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm [ underset{leftVert urightVert _{W^{1,n}left(mathbb{R} ^{n}right) }=1}{sup}int_{ mathbb{R}^{n}}Phileft( alpha_{n}leftvert urightvert ^{frac{n}{n-1}}left( 1+alphaleftVert urightVert _{n}^{n}right) ^{frac{1}{n-1}}right) dx<+infty ]in the entire space $mathbb{R}^n$ for any $0leqalpha<1$, where $Phileft( tright) =e^{t}-underset{j=0}{overset{n-2}{sum}}% frac{t^{j}}{j!}$, $alpha_{n}=nomega_{n-1}^{frac{1}{n-1}}$ and $omega_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $mathbb{R}^n$. We also show that the above supremum is infinity for all $alphageq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $alpha>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $Phi(t)$ is replaced by a strictly smaller $Phi^*(t)=e^{t}-underset{j=0}{overset{n-1}{sum}}% frac{t^{j}}{j!}$. (Note that $Phi(t)=Phi^*(t)+frac{t^{n-1}}{(n-1)!}$).