We prove the existence of extremal functions of Sobolev-Poincar\'e inequality on $S^n$ for $p\in(1,(1+\sqrt{1+8n})/4)$. For general $n$-dimensional compact Riemannian manifolds embedded in $R^{n+1}$, such an existence result is proved for $p \in (n/(n-1),(1+\sqrt{1+8n})/4)$.