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

We establish the Hopf boundary point lemma for the Schrödinger operator Δ + V involving potentials V that merely belong to the space Lloc1(Ω). More precisely, we prove that among all nonnegative supersolutions u of Δ + V which vanish on the boundary Ω and are such that V u L1(Ω), if there exists one supersolution that satisfies un < 0 almost everywhere on Ω 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(Ω).

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Hopf lemma, boundary point lemma, Schrödinger operator, weak normal derivative
Mathematical Subject Classification 2010
Primary: 35B05, 35B50
Secondary: 31B15, 31B35
Received: 3 June 2017
Accepted: 9 April 2018
Published: 6 June 2018
Luigi Orsina
Dipartimento di Matematica
“Sapienza” Università di Roma
Augusto C. Ponce
Institut de Recherche en Mathématique et Physique
Université catholique de Louvain