Strong maximum principle for Schrödinger operators with singular potential (Q252566)

This paper investigates the strong maximum principle for Schrödinger operators of the form \(-\Delta + V\). The main result is the following. If \(\Omega \subset \mathbb{R}^N\) is an open connected set, \(p > 1\), \(V \in L^p(\Omega)\), \(u\in L^1(\Omega)\) is a nonnegative function such that \(Vu \in L^1(\Omega)\) and \(-\Delta u + V u \geq 0\) in the sense of distributions in \(\Omega\), and if the average integral of \(u\) satisfies \(\lim_{r\to 0} \int_{B(x,r)} u = 0\) for every point \(x\) in a compact subset of \(\Omega\) with positive \(W^{2,p}\) capacity, then \(u=0\) almost everywhere in \(\Omega\).