Abstract
AbstractIt has been shown by Trudinger and Moser that for normalized functions u of the Sobolev space 𝕎1, N (Ω), where Ω is a bounded domain in ℝN, one has ∫Ω exp(αN|u|N/(N − 1))dx ≤ CN, where αN is an explicit constant depending only on N, and CN 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 ℝN. 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.
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