Strong maximum principle for Schrödinger operators with singular potential

We prove that for every p>1p>1 and for every potential V∈LpV∈Lp, any nonnegative function satisfying −Δu+Vu≥0−Δu+Vu≥0 in an open connected set of RNRN is either identically zero or its level set {u=0}{u=0} has zero W2,pW2,p capacity. This gives an affirmative answer to an open problem of Bénilan and Brezis concerning a bridge between Serrin–Stampacchia's strong maximum principle for p>N2 and Ancona's strong maximum principle for p=1p=1. The proof is based on the construction of suitable test functions depending on the level set {u=0}{u=0}, and on the existence of solutions of the Dirichlet problem for the Schrödinger operator with diffuse measure data.