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 +∞, while in the second the perturbation is (p−1)−superlinear near +∞ and in the third we do not require asymptotic condition at +∞. Using variational methods together with truncation and comparison techniques, we show that for λ∈(0,λˆ1) - λ>0 is the parameter and λˆ1 being the principal eigenvalue of (−Δp,W1,p0(Ω)) - we have positive solutions, while for λ≥λˆ1, 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 +∞, provided that the perturbation is damped by a parameter

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

CANDITO, Pasquale
;
2017

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 +∞, while in the second the perturbation is (p−1)−superlinear near +∞ and in the third we do not require asymptotic condition at +∞. Using variational methods together with truncation and comparison techniques, we show that for λ∈(0,λˆ1) - λ>0 is the parameter and λˆ1 being the principal eigenvalue of (−Δp,W1,p0(Ω)) - we have positive solutions, while for λ≥λˆ1, 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 +∞, provided that the perturbation is damped by a parameter
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12318/6730
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