A sharp Trudinger–Moser type inequality for unbounded domains in [formula omitted

Abstract

The classical Trudinger–Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H 0 1 ( Ω ) (with Ω ⊂ R 2 a bounded domain), the integral ∫ Ω e 4 π u 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 2 ). In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫ Ω e 4 π u 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 = B R , the ball or radius R, and for Ω = R 2 . Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls B R ⊂ R 2 and on R 2 .