Schrodinger equations with very singular potentials in Lipschitz domains

Consider operators L := ∆ + V in a bounded Lipschitz domain Ω ⊂ R . Assume that V ∈ C(Ω) and V satisfies V (x) ≤ ā dist (x, ∂Ω) in Ω and also a second condition that guarantees the existence of a ground state ΦV . If, for example, V > 0 this condition reads 1 < cH(V ) (= the Hardy constant relative to V ). We derive estimates of positive LV harmonic functions and of positive Green potentials of measures τ ∈ M+(Ω;ΦV ). These imply estimates of positive LV superharmonic functions and of LV subharmonic functions that are dominated by an LV superharmonic. Similar results have been obtained in [7] in the case of smooth domains. MSC:35J60; 35J75; 35J10