Electron. J. Diff. Eqns., Vol. 2007(2007), No. 87, pp. 1-5.

A note on extremal functions for sharp Sobolev inequalities

Ezequiel R. Barbosa, Marcos Montenegro

Abstract:
In this note we prove that any compact Riemannian manifold of dimension $n\geq 4$ which is non-conformal to the standard n-sphere and has positive Yamabe invariant admits infinitely many conformal metrics with nonconstant positive scalar curvature on which the classical sharp Sobolev inequalities admit extremal functions. In particular we show the existence of compact Riemannian manifolds with nonconstant positive scalar curvature for which extremal functions exist. Our proof is simple and bases on results of the best constants theory and Yamabe problem.

Submitted March 23, 2007. Published June 15, 2007.
Math Subject Classifications: 32Q10, 53C21.
Key Words: Extremal functions; optimal Sobolev inequalities; conformal deformations.

Show me the PDF file (172K), TEX file, and other files for this article.

Ezequiel R. Barbosa
Departamento de Matemática
Universidade Federal de Minas Gerais
Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
email: ezequiel@mat.ufmg.br
Marcos Montenegro
Departamento de Matemática
Universidade Federal de Minas Gerais
Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
email: montene@mat.ufmg.br

Return to the EJDE web page