A note on extremal functions for sharp Sobolev inequalities
Texas State University-San Marcos, Department of Mathematics
In this note we prove that any compact Riemannian manifold of dimension n ≥ 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.
Extremal functions, Optimal Sobolev inequalities, Conformal deformations
Barbosa, E., & Montenegro, M. (2007). A note on extremal functions for sharp Sobolev inequalities. <i>Electronic Journal of Differential Equations, 2007</i>(87), pp. 1-5.
Attribution 4.0 International