Concave-Convex Problems for the Robin p -Laplacian Plus an Indefinite Potential

Nikolaos S. Papageorgiou and Andrea Scapellato
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Nikolaos S. Papageorgiou: Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece
Andrea Scapellato: Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale Andrea Doria 6, 95125 Catania, Italy

Mathematics, 2020, vol. 8, issue 3, 1-27

Abstract: We consider nonlinear Robin problems driven by the p -Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, ( p − 1 ) -sublinear) term and of a convex (that is, ( p − 1 ) -superlinear) term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0 . In addition, we show the existence of a smallest positive solution u λ * and determine the monotonicity and continuity properties of the map λ ? u λ * .

Keywords: concave-convex nonlinearities; p -Laplacian; indefinite potential; antimaximum principle; nonlinear regularity; positive solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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