A sharp Trudinger-Moser type inequality for unbounded domains in ${\mathbb{R}}^n$

Abstract

The Trudinger-Moser inequality states that for functions $u\in H_0^{1,n}(\Omega )$ ($\Omega \subset {\mathbb{R}}^n$ a bounded domain) with ${\int }_{\Omega }|\nabla u{|}^{n}dx\le 1$ one has ${\int }_{\Omega }(e^{{\alpha }_n|u{|}^{\frac{n}{n-1}}}-1)dx\le c|\Omega |$, with $c$ independent of $u$. Recently, the second author has shown that for $n=2$ the bound $c|\Omega |$ may be replaced by a uniform constant $d$ independent of $\Omega$ if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring ${\int }_{\Omega }(|\nabla u{|}^n+|u{|}^n)dx\le 1$. We extend here this result to arbitrary dimensions $n>2$. Also, we prove that for $\Omega ={\mathbb{R}}^n$ the supremum of ${\int }_{{\mathbb{R}}^n}(e^{{\alpha }_n|u{|}^{\frac{n}{n-1}}}-1)dx$ over all such functions is attained. The proof is based on a blow-up procedure.