Abdelouahed El Khalil

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

We prove some properties of the first eigenvalue for the elliptic system −Δpu=λ|u|α|v|βv in Ω, −Δqv=λ|u|α|v|βu in Ω, (u,v)∈W01,p(Ω)×W01,q(Ω). In particular, the first eigenvalue is shown to be simple. Moreover, the stability with respect to (p,q) is established.

On the Principal Eigencurve of the p-Laplacian: Stability Phenomena

We show that each point of the principal eigencurve of the nonlinear problem −�pu − �m(x)|u| p 2 u = µ|u| p 2 u in ,

A nonlinear boundary problem involving the p-bilaplacian operator

We show some new Sobolevs trace embedding that we apply to prove that the fourth-order nonlinear boundary conditions Δp2u

p ( x ) p(x) -biharmonic operator involving the p ( x ) p(x) -Hardy inequality

Abstract In this work, we investigate the spectrum denoted by Λ for the p ⁢ ( x ) {p(x)} -biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup ⁡ Λ = + ∞ {\sup\Lambda=+\infty} . Moreover, we prove that inf ⁡ Λ > 0 {\inf\Lambda>0} if and only if the domain Ω satisfies the p ⁢ ( x ) {p(x)} -Hardy inequality.

On the -Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the -biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.

On a Class of PDE Involving -Biharmonic Operator

The existence of solution for a fourth-order nonlinear partial differential equation (PDE) class involving p-biharmonic operator Δ(|Δ𝑢|𝑝−2Δ𝑢)=𝜆𝜌(𝑥)|𝑢|𝑞−2𝑢inΩ,𝑢=Δ𝑢=0,on𝜕Ω, is proved by applying mountain pass theorem and a local minimization.