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Journal of Convex Analysis 26 (2019), No. 4, 1145--1174
Copyright Heldermann Verlag 2019

Positive Solutions for Nonlinear Robin Problems with Concave Terms

Leszek Gasinski
Dept. of Mathematics, Pedagogical University, 30-084 Cracow, Poland

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

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


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

Keywords: p-Laplacian, concave nonlinearity, uniform nonresonance, nonuniform nonresonance, bifurcation-type theorem, minimal positive solution.

MSC: 35J20, 35J60

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