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

Abstract

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