  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 $\mathbb{R}^N$ 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.
