Vol. 11, No. 8, 2018

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Hopf potentials for the Schrödinger operator

Luigi Orsina and Augusto C. Ponce

Vol. 11 (2018), No. 8, 2015–2047
DOI: 10.2140/apde.2018.11.2015
Abstract

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

Keywords
Hopf lemma, boundary point lemma, Schrödinger operator, weak normal derivative
Mathematical Subject Classification 2010
Primary: 35B05, 35B50
Secondary: 31B15, 31B35