Schrodinger equations with very singular potentials in Lipschitz domains

Consider operators $L^{V}:=Δ+ V$ in a bounded Lipschitz domain $Ω\subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(Ω)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partialΩ)^{-2}$ in $Ω$ and a second condition that guarantees the existence of a ground state $Φ_V$. 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 $τ\in {\mathfrak M}_+(Ω;Φ_V)$. 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.

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