A sharp Trudinger - Moser type inequality for unbounded domains in R

The classical Trudinger-Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H 0 (Ω) (with Ω ⊂ R a bounded domain), the integral ∫ Ω e 2 dx is uniformly bounded by a constant depending only on Ω. If the volume |Ω| becomes unbounded then this bound tends to infinity, and hence the Trudinger-Moser inequality is not available for such domains (and in particular for R). In this paper we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫ Ω e 2 dx over all such functions is uniformly bounded, independently of the domain Ω. Furthermore, a sharp upper bound for the limits of Sobolev normalized concentrating sequences is proved for Ω = BR, the ball or radius R, and for Ω = R. Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls BR ⊂ R and on R.