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

Abstract

The Trudinger-Moser inequality states that for functionswith 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., requiringWe extend here this result to arbitrary dimensions n > 2. Also, we prove that for Ω = R n the supremum of R n (e α n |u| n/(n-1) -1) dx over all such functions is attained.The proof is based on a blow-up procedure.