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

TL;DR

This paper investigates the conditions under which the strong maximum principle and Green's function exist for Schrödinger operators with nonnegative potentials, revealing that certain points prevent Green's function existence and affect solution properties.

Contribution

It establishes the nonexistence of Green's function at specific points and characterizes when solutions exist for Schrödinger equations with measure data.

Findings

Strong maximum principle holds in Sobolev-connected components outside a critical set Z.

Green's function does not exist at points in Z.

Solutions exist for measures supported outside Z.

Abstract

Given any Borel function $V : \Omega \to [0, +\infty]$ on a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schr\"odinger operator $-\Delta + V$ in $\Omega$ holds in each Sobolev-connected component of $\Omega \setminus Z$, where $Z \subset \Omega$ is the set of points which cannot carry a Green's function for $- \Delta + V$. More generally, we show that the equation $- \Delta u + V u = \mu$ has a distributional solution in $W_{0}^{1, 1}(\Omega)$ for a nonnegative finite Borel measure $\mu$ if and only if $\mu(Z) = 0$.