A sharp Trudinger–Moser type inequality for unbounded domains in ${\mathbb{R}}^2$

Abstract

The classical Trudinger–Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space $H_0^1(\Omega )$ (with $\Omega \subset {\mathbb{R}}^2$ a bounded domain), the integral ${\int }_{\Omega }{e}^{4\pi u^2}\mathit{dx}$ is uniformly bounded by a constant depending only on $\Omega$. If the volume $\left|\Omega \right|$ becomes unbounded then this bound tends to infinity, and hence the Trudinger–Moser inequality is not available for such domains (and in particular for ${\mathbb{R}}^2$).

In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ${\int }_{\Omega }{e}^{4\pi u^2}\mathit{dx}$ 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 $\Omega ={\mathbb{R}}^2$. Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls $B_R\subset {\mathbb{R}}^2$ and on ${\mathbb{R}}^2$.