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On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil;  Mohammed Ouanan — 2005

Applicationes Mathematicae

We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ Δ ₚ u = | u | p - 2 u in Ω, ⎨ ⎩ | ∇ u | p - 2 ∂ u / ∂ ν = λ ϱ ( x ) | u | p - 2 u + μ | u | p - 2 u on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

On the spectrum of the p-biharmonic operator involving p-Hardy's inequality

Abdelouahed El Khalil;  My Driss Morchid Alaoui;  Abdelfattah Touzani — 2014

Applicationes Mathematicae

In this paper, we study the spectrum for the following eigenvalue problem with the p-biharmonic operator involving the Hardy term: Δ ( | Δ u | p - 2 Δ u ) = λ ( | u | p - 2 u ) / ( δ ( x ) 2 p ) in Ω, u ∈ W ₀ 2 , p ( Ω ) . By using the variational technique and the Hardy-Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.

A Weighted Eigenvalue Problems Driven by both p ( · ) -Harmonic and p ( · ) -Biharmonic Operators

Mohamed Laghzal;  Abdelouahed El Khalil;  Abdelfattah Touzani — 2021

Communications in Mathematics

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p ( · ) -Harmonic and p ( · ) -biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u ∈ W 2 , p ( · ) ( Ω ) ∩ W 0 1 , p ( · ) ( Ω ) , is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces L p ( · ) ( Ω ) and W m , p ( · ) ( Ω ) .

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