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

  • * Corresponding author: P. Candito

    * Corresponding author: P. Candito 

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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  • 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.

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


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