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On an inequality by N. Trudinger and J. Moser and related elliptic equations
Author(s) -
de Figueiredo Djairo G.,
Marcos do Ó João,
Ruf Bernhard
Publication year - 2002
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10015
Subject(s) - mathematics , bounded function , unit sphere , sobolev space , domain (mathematical analysis) , constant (computer programming) , interpretation (philosophy) , pure mathematics , ball (mathematics) , sequence (biology) , generalization , mathematical analysis , bounded mean oscillation , biology , computer science , genetics , programming language
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.