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Journal of Convex Analysis 24 (2017), No. 1, 261--285
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, 30-348 Kraków, Poland

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


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, p-logistic equation.

MSC: 35J20, 35J65, 35P30

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