On an inequality by N. Trudinger and J. Moser and related elliptic equations

It has been shown by Trudinger and Moser that for normalized functions $u$ of the Sobolev space $\mathbb{W}^{1,N}(\Omega)$, where $\Omega$ is a domain in $\mathbb{R}^{N},$ the integral $\int_{\Omega}\exp(u^{\alpha_{N}N/(N-1)})dx$ remains uniformly bounded. Carleson and Chang proved that there exists a corresponding extremal function in the case that $\Omega$ is the unit ball in $\mathbb{R}^{N}.$ In this paper we give a new proof, a generalization, and a new interpretation of this result. In particular, we give an explicit sequence which is maximizing for the above integral among all normalized ''concentrating sequences ''.