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 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...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 2002-02, Vol.55 (2), p.135-152
Main Authors: de Figueiredo, Djairo G., Marcos do Ó, João, Ruf, Bernhard
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Real functions
Sciences and techniques of general use
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Summary:It 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.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.10015