Hopf potentials for the Schrödinger operator

Abstract

We establish the Hopf boundary point lemma for the Schrödinger operator $-\mathrm{\Delta }+V$ involving potentials $V$ that merely belong to the space $L_{loc}^1\left(\mathrm{\Omega }\right)$. More precisely, we prove that among all nonnegative supersolutions $u$ of $-\mathrm{\Delta }+V$ which vanish on the boundary $\partial \mathrm{\Omega }$ and are such that $Vu\in L^1\left(\mathrm{\Omega }\right)$, if there exists one supersolution that satisfies $\partial u∕\partial n<0$ almost everywhere on $\partial \mathrm{\Omega }$ with respect to the outward unit vector $n$, then such a property holds for every nontrivial supersolution in the same class. We rely on the existence of nontrivial solutions of the nonhomogeneous Dirichlet problem with boundary datum in $L^{\infty }\left(\partial \mathrm{\Omega }\right)$.