Abstract

We consider a parametric Robin problem driven by the pp-Laplacian plus a potential. In the reaction we have the combined effects of a parametric concave term and of a (p ⁣− ⁣1)(p \!-\! 1)-linear perturbation. We consider the case of uniform nonresonance with respect to the principal eigenvalue λ^1>0\widehat{\lambda}_1>0 and the case of nonuniform nonresonance with respect to λ^1>0\widehat{\lambda}_1>0. For both cases we prove a bifurcation-type theorem describing the dependence on the parameter λ>0\lambda>0 of the set of positive solutions. We also establish the existence of a smallest positive solution u~λ∗\widetilde{u}^*_{\lambda} for every admissible parameter λ>0\lambda>0 and determine the monotonicity and continuity properties of the map λ⟼u~λ∗\lambda\longmapsto\widetilde{u}_{\lambda}^*.

Author information

Contact details are reproduced from the original publication and may be historical.

Nikolaos S. Papageorgiou

Dept. of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

npapg@math.ntua.gr

Krzysztof Winowski

Fac. of Mathematics and Computer Science, Jagiellonian University, 30-348 Cracow, Poland

Keywords

  • p-Laplacian
  • concave nonlinearity
  • uniform nonresonance
  • nonuniform nonresonance
  • bifurcation-type theorem
  • minimal positive solution

Mathematics Subject Classification

35J20, 35J60

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Suggested citation

L. Gasiński, N. S. Papageorgiou, K. Winowski. “Positive Solutions for Nonlinear Robin Problems with Concave Terms.” Journal of Convex Analysis 26 (2019), No. 4, 1145–1174. https://doi.org/10.68381/jca26063

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.