Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems

Gabriele Bonanno; Pasquale Candito; Roberto Livrea; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2017057

Article Contents

2017, Volume 16, Issue 4: 1169-1188. Doi: 10.3934/cpaa.2017057

This issue Previous Article Nonlinear Dirichlet problems with double resonance Next Article A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space

Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems

1.
Department of Engineering, University of Messina, Messina, 98166, Italy
2.
Department DICEAM, University of Reggio Calabria, Reggio Calabria, 89122, Italy
3.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

^* Corresponding author: P. Candito
^* Corresponding author: P. Candito

Received: April 30, 2016

Revised: January 31, 2017

Published: May 2017

The authors have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet $p-$Laplacian. We consider three cases. In the first the perturbation is $(p-1)-$sublinear near $+\infty$, while in the second the perturbation is $(p-1)-$superlinear near $+\infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in (0, \widehat{\lambda}_1)$ -$\lambda>0$ is the parameter and $\widehat{\lambda}_1$ being the principal eigenvalue of $\left(-\Delta_p, W^{1, p}_0(\Omega)\right)$ -we have positive solutions, while for $\lambda\geq \widehat{\lambda}_1$, no positive solutions exist. In the "sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 35J80.

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