Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 87, pp. 1-5.

Title: A note on extremal functions for sharp Sobolev
       inequalities

Authors: Ezequiel R. Barbosa (Univ. Federal de Minas Gerais, Brazil)
         Marcos Montenegro   (Univ. Federal de Minas Gerais, Brazil)

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.