Consider the equation
$-\Delta u + Vu=f$,
on a closed manifold (or on a bounded domain with homogeneous Neumann condition). Here one can assume whatever integrability or smoothness conditions on $V$ and $f$ one likes. One can show that if $V$ and $f$ are both nonnegative and not identically zero, then the unique solution $u$ is strictly positive. Moreover, $u$ is bounded from below by a constant $C$ times an appropriate norm of $f$, where $C$ does not depend on $f$. My question is, where or how can I find more information on how $C$ depends on $V$?