On the extremal functions of Sobolev-Poincaré inequality

Abstract

We prove the existence of extremal functions of Sobolev-Poincaré inequality on Sn for p (1, (1 + √1 + 8n)/4). For general n-dimensional compact Riemannian manifolds embedded in Rn+1, such an existence result is proved for p ∈ (n/(n-1), (1 + √1 + 8n)/4).