Can we take $C_0=0$ in abobe theorem? Because I just need the Euclidean type : $||u||_q\le C_1||\nabla u||_p where $$ u\in C_0^{\infty}(D) $ $ and $$\frac{1}{p}=\frac{1}{q}-\frac{1}{n}.$

Dear Ribeiro, yes my question has a typo." Let $M$ be a complete non-compact Riemannian manifold. Let $D$ be a bounded domain with smooth boundary $\partial D$ in $M$. What is the minimum requirement (about domain 𝐷, curvature,..) so that the Euclidean type 𝐿𝑝 Sobolev inequality holds on 𝐷?" You are saying that Euclidean type $L^p$ -Sobolev inequality holds on a bounded domain $D$ with smooth boundary $\partial D$ in $M$. Can you give me a reference please? Thanks for your answer.