We prove that for every and for every potential , any nonnegative function satisfying in an open connected set of is either identically zero or its level set has zero 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 and Ancona's strong maximum principle for . The proof is based on the construction of suitable test functions depending on the level set , 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. 477–493DOI 10.1016/J.ANIHPC.2014.11.004