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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, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ 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, $u \in W ₀^{2, p} (Ω)$ . 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.

On the p-biharmonic operator with critical Sobolev exponent

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

Applicationes Mathematicae

We study the existence of solutions for a p-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a nontrivial solution.

A Weighted Eigenvalue Problems Driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -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 $p (\cdot)$ -Harmonic and $p (\cdot)$ -biharmonic operators $\begin{matrix} Δ_{p (x)}^{2} u - Δ_{p (x)} u = λ w (x) {| u |}^{q (x) - 2} u in Ω, \\ u \in W^{2, p (\cdot)} (Ω) \cap W_{0}^{1, p (\cdot)} (Ω), \end{matrix}$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p (\cdot)} (Ω)$ and $W^{m, p (\cdot)} (Ω)$ .

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Currently displaying 1 – 12 of 12

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

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

On the p-biharmonic operator with critical Sobolev exponent

A Weighted Eigenvalue Problems Driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -Biharmonic Operators

A global bifurcation result of a Neumann problem with indefinite weight.

Boundary eigencurve problems involving the $p$ -Laplacian operator.

Existence and regularity of positive solutions for an elliptic system.

Bifurcation of nonlinear elliptic system from the first eigenvalue.

On the spectrum of the $p$ -biharmonic operator.

Stability of the principal eigenvalue of a nonlinear elliptic system.

Simplicity and stability of the first eigenvalue of a nonlinear elliptic system.

A nonlinear boundary problem involving the $p$ -bi-Laplacian operator.