Strong maximum principle for Schr\"odinger operators with singular potential

We prove that for every $p>1$ and for every potential $V\in L^p$, any nonnegative function satisfying $-\mathrm{\Delta }u+Vu\ge 0$ in an open connected set of ${\mathbb{R}}^N$ 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 B'enilan and Brezis concerning a bridge between Serrin-Stampacchia’s strong maximum principle for $p>\frac{N}{2}$ 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 Schr"odinger operator with diffuse measure data.