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 $V(x)\le \overline{a}\mathrm{dist}(x,\partial \Omega {)}^{-2}$ 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.

18 pages