Hopf potentials for the Schrödinger operator

Abstract

We establish the Hopf boundary point lemma for the Schrödinger operator $-Δ+ V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(Ω)$. More precisely, we prove that among all supersolutions $u$ of $-Δ+ V$ which vanish on the boundary $\partialΩ$ and are such that $V u \in L^{1}(Ω)$, if there exists one supersolution which satisfies $\partial u/\partial n < 0$ almost everywhere on $\partialΩ$ 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 $L^{\infty}(\partialΩ)$.