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 $V$?



Lower bound on the solution of a Schrödinger-type equation
Here are a couple of ideas:


1 This yields a lower bound, but it is not in terms of a norm of f:


Let $\lambda=\min(f/V)$. Then $u-\lambda$ satisifies
$$-\Delta(u-\lambda)+V(u-\lambda)=f-\lambda V\ge 0.$$
Hence $u\ge\lambda$ by the maximum principle.


2 Assume V is bounded, and let $V\_M$ be its maximum:
Let $v$ be the solution of $-\Delta v+V\_Mv=f$. Then
$$-\Delta(u-v)+V(u-v)=(V\_M-V)v\ge 0,$$
so $u\ge v$. If the geometry of the domain is simple, you may
be able to determine the Green's function for $-\Delta+V\_M$ explicitly.


