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

Abstract

Let $W^{1,n}({\mathbb{R}}^n$ be the standard Sobolev space and ${\Vert \cdot \Vert }_n$ 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 \[ \underset{\left\Vert u\right\Vert _{W^{1,n}\left(\mathbb{R} ^{n}\right) }=1}{\sup}\int_{ \mathbb{R}^{n}}Φ\left( α_{n}\left\vert u\right\vert ^{\frac{n}{n-1}}\left( 1+α\left\Vert u\right\Vert _{n}^{n}\right) ^{\frac{1}{n-1}}\right) dx<+\infty \]in the entire space ${\mathbb{R}}^n$ for any $0\le \alpha <1$, where $\Phi \left(t\right)=e^t-\underset{j=0}{\sum ^{n-2}}$, ${\alpha }_n=n{\omega }_{n-1}^{\frac{1}{n-1}}$ 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 $\alpha \ge 1$. 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 ${\Phi }^{\ast }(t)=e^t-\underset{j=0}{\sum ^{n-1}}$. (Note that $\Phi (t)={\Phi }^{\ast }(t)+\frac{t^{n-1}}{(n-1)!}$).

33 pages, submitted for publication on February 8, 2017

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