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 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).

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