
Journal of Convex Analysis 26 (2019), No. 4, 11451174 Copyright Heldermann Verlag 2019 Positive Solutions for Nonlinear Robin Problems with Concave Terms Leszek Gasinski Dept. of Mathematics, Pedagogical University, 30084 Cracow, Poland leszek.gasinski@up.krakow.pl 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, 30348 Cracow, Poland [Abstractpdf] 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 bifurcationtype 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: pLaplacian, concave nonlinearity, uniform nonresonance, nonuniform nonresonance, bifurcationtype theorem, minimal positive solution. MSC: 35J20, 35J60 [ Fulltextpdf (202 KB)] for subscribers only. 