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

Authors: Yuxiang Li and Bernhard Ruf

Issue: Volume 57 (2008), Issue 1, 451-480

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 $\int_{\Omega} |\nabla u|^n \mathrm{d}x \le 1$, one has \[ \int_{\Omega} (e^{\alpha_n |u|^{n/(n-1)}} - 1) \mathrm{d}x \le c|\Omega|, \] 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 \[ \int_{\Omega}(|\nabla u|^n + |u|^n) \mathrm{d}x \le 1. \] We extend here this result to arbitrary dimensions $n > 2$. Also, we prove that for $\Omega = \mathbb{R}^n$ the supremum of $\int_{\mathbb{R}^n} (e^{\alpha_n|u|^{n/(n-1)}} - 1) \mathrm{d}x$ over all such functions is attained. The proof is based on a blow-up procedure. (See a graphic rendition of this abstract in http://www.iumj.indiana.edu/oai/2008/57/3137/3137_abstract.xml.)