I am trying to find some reference/proofs regarding $L^p$ regularity theory for elliptic pdes on the whole space (or more generally on unbounded domains and/or domains with infinite measure). The basic thing I want to understand is under what hypotheses we can say that if $u\in H^{1}(\mathbb{R}^n)$ satisfies:
$$- \Delta u + c(x) u= f$$
with $f \in L^p$ with $p >1$ and $c(x)$ a positive continuous function (possibly Holder continuous, or even bounded, I don't care about being as general as possible) then $ \nabla u \in L^{p^*}$. I've managed to find only local estimates, not dealing with the whole space. Of course the elliptic operator can be generalised, some conditions on $p$ might be needed etc., I just would like to see some results regarding $L^p$ regularity on the whole space.
Do you know references/results that give global (and not only local) regularity?