Geodesic
A note on extremal functions for sharp Sobolev inequalities
Electronic journal of differential equations, Tome 2007 (2007)
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.
Classification : 32Q10, 53C21
Keywords: extremal functions, optimal Sobolev inequalities, conformal deformations 
@article{EJDE_2007__2007__a14,
     author = {Barbosa,  Ezequiel R. and Montenegro,  Marcos},
     title = {A note on extremal functions for sharp {Sobolev} inequalities},
     journal = {Electronic journal of differential equations},
     year = {2007},
     volume = {2007},
     zbl = {1135.53024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a14/}
}
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%J Electronic journal of differential equations
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%F EJDE_2007__2007__a14
Barbosa,  Ezequiel R.; Montenegro,  Marcos. A note on extremal functions for sharp Sobolev inequalities. Electronic journal of differential equations, Tome 2007 (2007). http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a14/