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 1, N (Ω), where Ω is a bounded domain in ℝ N , one has ∫ Ω exp(α N | u | N /( N − 1) ) dx ≤ C N , where α N is an explicit constant depending only on N , and C N is a constant depending only on N and Ω. Carleson and Chang proved that there exists a corresponding extremal function in the case that Ω is the unit ball in 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 that is maximizing for the above integral among all normalized “concentrating sequences.” As an application, the existence of a nontrivial solution for a related elliptic equation with “Trudinger‐Moser” growth is proved. © 2002 John Wiley & Sons, Inc.