On the nonexistence of Green's function and failure of the strong maximum principle

Given any Borel function $V:\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ on a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, we establish that the strong maximum principle for the Schr"odinger operator $-\mathrm{\Delta }+V$ in $\mathrm{\Omega }$ holds in each Sobolev-connected component of $\mathrm{\Omega }\setminus Z$, where $Z \subset \Omega$isthesetofpointswhichcannotcarryaGree{n}^{\mathrm{\prime }}sfunctionfor$- \Delta + V$\mathrm{.}Moregenerally,weshowthattheequation$- \Delta u + V u = \mu$hasadistributionalsolutionin$W_{0}^{1, 1}(\Omega)$ for a nonnegative finite Borel measure $\mu$ if and only if $\mu (Z)=0$.