Let $M$ be a compact manifold without boundary which is not orientable. Do all the standard facts that apply to oriented manifolds and Sobolev spaces also apply here? Like Green's formula for example.
In Hebey's books on this subject, he never says that the manifold is orientable, but this may be an implicit assumption because it seems that he uses the volume form on the manifold and doesn't say anything about densities...
Statements like Green's theorem are identities that are linear in the function(s) involved. This allows us to localize their consideration: cover $M$ by coordinate patches (open subsets diffeomorphic to $\mathbb{R}^n$). Let $\{\varphi_k\}$ be a smooth partition of unity subordinate to this cover. Sine $u=\sum_k u\varphi_k$, you apply the identity to $u \varphi_k$ (which holds since there are no issues with orientability locally), and add the results.