A sharp Trudinger-Moser type inequality for unbounded domains in R-2

The classical Trudinger-Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H-0(1)(Omega) (with Omega subset of R-2 a bounded domain), the integral integral(Omega)e(4piu2) dx is uniformly bounded by a constant depending only on Omega. If the volume \Omega\ 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-2). In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of integral(Omega)e(4piu2) dx over all such functions is uniformly bounded, independently of the domain Omega. Furthermore, a sharp upper bound for the limits of Sobolev normalized concentrating sequences is proved for Omega = B-R, the ball or radius R, and for Omega = R-2. Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls B-R subset of R-2 and on R-2.