A sharp Trudinger-Moser type inequality for unbounded domains in R2

In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫Ωe4πu2dx 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=R2. Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls BR⊂R2 and on R2.