Strong maximum principle for Schrödinger operators with singular potential

Ponce, Augusto [UCL] Orsina, Luigi

We prove that for every (p > 1) and for every potential (V in L^p), any nonnegative function satisfying (-Delta u + V u ge 0) in an open connected set of (R^N) is either identically zero or its level set ({u = 0}) has zero (W^{2, 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 > rac{N}{2}) and Ancona's strong maximum principle for (p = 1). The proof is based on the construction of suitable test functions depending on the level set ({u = 0}), and on the existence of solutions of the Dirichlet problem for the Schrödinger operator with diffuse measure data.