Euclidean type Sobolev inequality on Riemannian manifolds 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 $L^p$ Sobolev inequality holds on $D$?
 A: Your hypotheses are enough. More precisely, if $(M,g)$ is a complete Riemannian manifold (be it compact or not), one is guaranteed that a closed and bounded subset of $M$ is compact by the Hopf-Rinow theorem. This is the only role of completeness. In that case, one has that $\overline D\doteq D\cup\partial D$ is a compact submanifold of $M$ with smooth boundary $\partial D$ and nonvoid interior $\mathring{D}\doteq D\smallsetminus\partial D$.
Compactness is important because it allows us to get the Sobolev inequalities without any hypotheses on curvature or injectivity radius (more precisely, such bounds are automatic on $\overline{D}$ by compactness) by just passing to a finite open cover of $\overline{D}$ by coordinate chart domains in $M$ by means of a subordinate partition of unity. In fact, the following form of the $L^p$ Sobolev inequality holds:

Theorem: Let $n=\dim M\geq 2$, $1\leq q<n$ and $\frac{1}{p}=\frac{1}{q}-\frac{1}{n}(>0)$. Then for all $u\in\mathscr{C}^\infty(\overline{D})$ we have that $$\|u\|_p\leq C_1\|\nabla u\|_q+C_0\|u\|_q$$ for some $C_0,C_1>0$.

In fact, it suffices to prove the Theorem for $q=1$, that is, $p=\frac{n}{n-1}$.
All other $L^p$ Sobolev inequalities involving higher order derivatives and non-endpoint choices of $1\leq p,q<\infty$ follow by iterating the above inequality and applying Hölder's inequality, just like in the Euclidean case (edit: the Euclidean form of the Sobolev inequalities - that is, with $C_0=0$ - needs more work, see the end of the answer below)
If $u\in\mathscr{C}^\infty_c(\mathring{D})\subset\mathscr{C}^\infty(\overline{D})$, the proof is easily reducible to the Euclidean case by means of a partition of unity - more precisely, let $(U_1,\psi_1),\ldots,(U_m,\psi_m)$ be coordinate charts on $M$ adapted to $\partial D$ whenever $U_j\cap\partial D\neq\varnothing$ such that $\{U_1,\ldots,U_m\}$ is an open cover of $\overline{D}$, and let $\{f_1,\ldots,f_m\}$ be a partition of unity of $\overline{D}$ subordinate to this cover (i.e. $f_j\in\mathscr{C}^\infty_c(U_j)$, $0\leq f_j\leq 1$, $f_1(p)+\cdots+f_m(p)=1$ for all $p\in\overline{D}$). Then to get the above theorem is suffices to prove for all $j=1,\ldots,m$ the inequalities $$\|f_ju\|_p\leq K_j(\|f_j u\|_q+\|\nabla(f_j u)\|_q)\ ,\quad K_j>0\ ,$$ which on their turn clearly follow from the Euclidean $L^p$ Sobolev inequalities together with the fact that on $\mathrm{supp} f_j\subset U_j$ the components of $(\psi_j^{-1})^*g$ together with all their derivatives are bounded for each $j$.
Indeed, since $|\nabla(f_j u)|\leq|\nabla u|+|\nabla f_j||u|$ ($|\nabla u|\doteq\sqrt{g(\nabla u,\nabla u)}$), we have that $$\|u\|_p\leq\sum^m_{j=1}K_j(\|f_j u\|_q+\|\nabla(f_j u)\|_q)\leq m\left(\sup_{1\leq j\leq m}K_j\right)\left[\|\nabla u\|_q+\left(1+\sup_{1\leq j\leq m}\|\nabla f_j\|_\infty\right)\|u\|_q\right]\ .$$ In fact, the above proof works for every $u\in\mathscr{C}^\infty_c(M)$ (the constants will depend on $\mathrm{supp}\,u$, of course). The details of the above argument may be found in the proofs of Theorem 2.20, pp. 44-45 and Theorem 2.30, pp. 50-53 of the excellent book by the late Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry (Springer-Verlag, 1998), which also proves the $L^\infty$ version of the Sobolev inequalities and dedicates a considerable space to the determination of their optimal constants, a topic in which Aubin was a pioneer.
The general case $u\in\mathscr{C}^\infty(\overline{D})$ requires us to deal with the case that $\mathrm{supp}\,u\cap\partial D\neq\varnothing$. Unfortunately, Aubin's book does not provide all details on how to do this, which is usually done by appealing to a so-called smooth extension operator - that is, a linear map $\mathsf{K}:\mathscr{C}^\infty(\overline{D})\rightarrow\mathscr{C}^\infty_c(M)$ which is bounded with respect to all $L^p$ Sobolev norms, $1\leq p\leq\infty$ and such that $(\mathsf{K}u)|_{\overline{D}}=u$. It then suffices to apply the above proof to $\mathsf{K}u$ and finish the argument using the properties of $\mathsf{K}$.
Usually one requires a minimum of regularity from $\partial D$ in order to build such a $\mathsf{K}$ - typically one requires at least that $\partial D$ be Lipschitz (there are more specialized constructions for rougher boundaries, which I will not consider here). If $\partial D$ is smooth as you assumed, there is a particularly simple construction of such a $\mathsf{K}$ due to Seeley (Extensions of $C^\infty$ Functions Defined in a Half Space, Proc. Amer. Math. Soc. 15 (1963) 625-626). As the article's name suggests, Seeley actually did it for a closed half-space in $\mathbb{R}^n$ instead of $\overline{D}$, but this can be applied to $(\psi_j^{-1})^*(f_j u)$ for each $j$ such that $\mathrm{supp}\,f_j\cap\partial D\neq\varnothing$ while ensuring that the extension is still compactly supported in $\psi_j(U_j)\subset\mathbb{R}^n$ and hence can be pulled back to $U_j$. I do not recall right now a textbook which constructs Seeley's extension operator as I sketched above, but the missing steps are not difficult to fill if you use Seeley's original result as a black box. In any case, I strongly recommend having a look at Aubin's book.
Edit: to show that we can set $C_0=0$ in the above Theorem, one needs to apply to the case $q=1$, $p=\frac{n}{n-1}$ the following results, due to Federer, Fleming and Santalò:

Theorem 1 (Federer-Fleming): given an $n$-dimensional Riemannian manifold $(M,g)$, $n\geq 2$, set $$S_\nu(M)=\inf_{f\in\mathscr{C}^\infty_c(M)}\frac{\|\nabla f\|_1}{\|f\|_{\frac{\nu}{\nu-1}}}$$ and $$I_\nu(M)=\inf_D\frac{\mathrm{Area}(\partial D)}{\mathrm{Volume}(D)^{\frac{\nu-1}{\nu}}}\ ,$$ $\nu\in(1,\infty]$, where the last infimum is taken along all bounded subsets $D$ of $M$ with nonvoid interior and smooth boundary. Then $S_\nu(M)=I_\nu(M)$ for all $\nu$.
Theorem 2 (Santalò): Let $I_\nu$ as above. If $D$ is a nonvoid, bounded open subset of the complete Riemannian manifold $(M,g)$ with smooth boundary $\partial D$, then $I_\infty(D)>0$.

(for a proof of Theorem 1 see e.g. Theorem VIII.3.2, pp. 363-366 of the book by I. Chavel, Riemannian Geometry - A Modern Introduction, Second Edition, Cambridge University Press, 2006. As for a proof of Theorem 2, see e.g. Theorem VIII.3.5, pp. 369 of I. Chavel, loc. cit.)
Both Theorems together imply that $\|u\|_1\leq \frac{1}{I_\infty(D)}\|\nabla u\|_1$ for all $u\in\mathscr{C}^\infty_c(\mathring{D})$, hence yielding the Euclidean Sobolev inequality (with a possibly different constant from that of Euclidean space) in the case $q=1$, $p=\frac{n}{n-1}$. The remainder cases with $q>1$ follow from the Hölder inequality and by noticing that for $u$ smooth (in fact, even for $u$ locally Lipschitz) we have that $|\nabla u|=|\nabla|u||$ almost everywhere (see e.g. Lemma VIII.3.2, pp. 365-366 of I. Chavel, loc. cit.). Unfortunately the Euclidean Sobolev inequality no longer necessarily holds if $\mathrm{supp}\,u\cap\partial D\neq\varnothing$ (take e.g. $u\equiv$ a nonzero constant in $\overline{D}$) - the next best thing that survives is the so-called Sobolev-Poincaré inequality: if $u\in\mathscr{C}^\infty(\overline{D})$, then for $p,q$ as in the above Theorem there is some $C>0$ such that $$\|u-\overline{u}\|_p\leq C\|\nabla u\|_q\ ,\quad\overline{u}=\frac{\int_{\overline{D}} u d\mu_g}{\int_{\overline{D}}d\mu_g} \,$$ where $d\mu_g$ is the Riemannian volume element of $(M,g)$.
