Hopf potentials for the Schr\"odinger operator

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