In several sources (Choquet-Bruhat & Christodoulou 1981, Nirenberg-Walker 1973) estimates for elliptic partial differential equations on a noncompact manifold are derived in weighted Sobolev spaces. These usually go along the lines: if $L$ is an elliptic operator of order $m$, then $$ \|u\|_{H_{s,\delta}} \leq C(\|Lu\|_{H_{s - m, \delta + m}} + \|u\|_{H_{0,\delta}}). $$ My questions is: is such an estimate type valid in the case of "standard" (non-weighted) Sobolev spaces $H^s$?

Edit: The manifold in question, though noncompact, may be assumed to have various convenient properties at infinity (for instance, asymptotically flat, bounds on the volume growth, etc.).