Schrodinger equations with very singular potentials in Lipschitz domains

Abstract

Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second condition that guarantees the existence of a ground state $\Phi_V$. If, for example, $V>0$ this condition reads $1<c_H(V)$ (= the Hardy constant relative to $V$). We derive estimates of positive $L_V$ harmonic functions and of positive Green potentials of measures $\tau\in {\mathfrak M}_+(\Omega;\Phi_V)$. These imply estimates of positive $L_V$ supersolutions and of $L_V$ subsolutions. Similar results have been obtained in [7] in the case of smooth domains.