
Journal of Convex Analysis 24 (2017), No. 1, 261285 Copyright Heldermann Verlag 2017 Positive, Extremal and Nodal Solutions for Nonlinear Parametric Problems Leszek Gasinski Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Kraków, Poland Leszek.Gasinski@ii.uj.edu.pl Nikolaos S. Papageorgiou National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece npapg@math.ntua.gr [Abstractpdf] We consider a nonlinear parametric problem driven by the $p$Laplace differential operator. For all large enough values of the parameter $\lambda$, we show that the problem has a smallest positive solution $u_{\lambda}^*\in C^1_0 (\overline{\Omega})$. We examine the monotonicity and continuity properties of the map $\lambda\longmapsto u^*_{\lambda}$. Finally we establish the existence of nodal (sign changing) solutions. Keywords: Extremal positive solutions, nonlinear regularity, nonlinear maximum principle, nodal solutions, plogistic equation. MSC: 35J20, 35J65, 35P30 [ Fulltextpdf (211 KB)] for subscribers only. 