%0 Journal Article
%T Strong maximum principle for Schr£żdinger operators with singular potential
%A Luigi Orsina
%A Augusto C. Ponce
%J Mathematics
%D 2013
%I arXiv
%X 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.
%U http://arxiv.org/abs/1311.4856v2