# Strong maximum principle for Schrödinger operators with singular potential

### Luigi Orsina

“Sapienza”, Università di Roma, Dipartimento di Matematica, P.le A. Moro 2, 00185 Roma, Italy### Augusto C. Ponce

Université catholique de Louvain, Institut de recherche en mathématique et physique, Chemin du cyclotron 2, L7.01.02, 1348 Louvain-la-Neuve, Belgium

## Abstract

We prove that for every $p > 1$ and for every potential $V \in L^{p}$, any nonnegative function satisfying $−\mathrm{\Delta }u + Vu \geq 0$ 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 Bénilan 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 Schrödinger operator with diffuse measure data.

## Cite this article

Luigi Orsina, Augusto C. Ponce, Strong maximum principle for Schrödinger operators with singular potential. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 477–493

DOI 10.1016/J.ANIHPC.2014.11.004