Abdel Rachid - Academia.edu (original) (raw)

Papers by Abdel Rachid

Abstract. In this work we study the existence of solutions for the nonlinear eigenvalue problem w... more Abstract. In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biharmonic ∆ 2 p u = λm(x)|u|p−2 u in a smooth bounded domain under Neumann boundary conditions. 1.

Applied Mathematics and Computation, 2011

Combining the minimax arguments and the Morse Theory, by computing the critical groups at zero, w... more Combining the minimax arguments and the Morse Theory, by computing the critical groups at zero, we establish the existence of a nontrivial solution for a class of Dirichlet boundary value problems, with resonance at infinity and zero. Résumé. Par un procédé de minimax et application de la Théorie de Morse, en calculant les groupes critiques en zéro, nousétablissons l'existence d'une solution non triviale pour une classe de problèmes de Dirichlet, avec résonanceà l'infini et en zéro..

In this article, we consider the nonlinear eigenvalue problem ∆(|∆u|p(x)−2∆u) = λ|u|q(x)−2u in Ω,... more In this article, we consider the nonlinear eigenvalue problem ∆(|∆u|p(x)−2∆u) = λ|u|q(x)−2u in Ω, u = ∆u = 0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary and p, q : Ω → (1, +∞) are continuous functions. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekelands variational principle.

Electronic Journal of Differential Equations

In this article we study the nonlinear Steklov boundary-value problem Δp(x)u = |u| p(x)-2u in Ω, ... more In this article we study the nonlinear Steklov boundary-value problem Δp(x)u = |u| p(x)-2u in Ω, Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.

Boletim da Sociedade Paranaense de Matemática, 2015

In this article, we study the following (p(x),q(x))(p(x),q(x))(p(x),q(x))-biharmonic type system \begin{gather*} \Del... more In this article, we study the following (p(x),q(x))(p(x),q(x))(p(x),q(x))-biharmonic type system \begin{gather*} \Delta(|\Delta u|^{p(x)-2}\Delta u)=\lambda F_u(x,u,v)\quad\text{in }\Omega,\\ \Delta(|\Delta v|^{q(x)-2}\Delta v)=\lambda F_v(x,u,v)\quad\text{in }\Omega,\\ u=v=\Delta u=\Delta v=0\quad \text{on }\partial\Omega. \end{gather*} We prove the existence of infinitely many solutions of the problem byapplying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

Electronic Journal of Qualitative Theory of Differential Equations, 2014

In this article, we study the nonlinear Steklov boundary-value problem ∆ p(x) u = |u| p(x)−2 u in... more In this article, we study the nonlinear Steklov boundary-value problem ∆ p(x) u = |u| p(x)−2 u in Ω, |∇u| p(x)−2 ∂u ∂ν = f (x, u) on ∂Ω. We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biha... more In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biharmonic ∆ 2 p u = λm(x)|u| p−2 u in a smooth bounded domain under Neumann boundary conditions.

In this note, we establish a variant of Ekeland's variational principle. This result suggests... more In this note, we establish a variant of Ekeland's variational principle. This result suggests a generalization of the classical Palais-Smale condition. An example is provided showing how this is used to give the existence of a minimizer for functionals which do not satisfy the Palais-Smale condition and the one introduced by Cerami. We also prove a relation between the coercitivity of functional and the introduced compactness condition.

Abstract. In this work we study the existence of solutions for the nonlinear eigenvalue problem w... more Abstract. In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biharmonic ∆ 2 p u = λm(x)|u|p−2 u in a smooth bounded domain under Neumann boundary conditions. 1.

Applied Mathematics and Computation, 2011

Combining the minimax arguments and the Morse Theory, by computing the critical groups at zero, w... more Combining the minimax arguments and the Morse Theory, by computing the critical groups at zero, we establish the existence of a nontrivial solution for a class of Dirichlet boundary value problems, with resonance at infinity and zero. Résumé. Par un procédé de minimax et application de la Théorie de Morse, en calculant les groupes critiques en zéro, nousétablissons l'existence d'une solution non triviale pour une classe de problèmes de Dirichlet, avec résonanceà l'infini et en zéro..

In this article, we consider the nonlinear eigenvalue problem ∆(|∆u|p(x)−2∆u) = λ|u|q(x)−2u in Ω,... more In this article, we consider the nonlinear eigenvalue problem ∆(|∆u|p(x)−2∆u) = λ|u|q(x)−2u in Ω, u = ∆u = 0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary and p, q : Ω → (1, +∞) are continuous functions. Considering different situations concerning the growth rates involved in the above quoted problem, we prove the existence of a continuous family of eigenvalues. The proofs of the main results are based on the mountain pass lemma and Ekelands variational principle.

Electronic Journal of Differential Equations

In this article we study the nonlinear Steklov boundary-value problem Δp(x)u = |u| p(x)-2u in Ω, ... more In this article we study the nonlinear Steklov boundary-value problem Δp(x)u = |u| p(x)-2u in Ω, Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.

Boletim da Sociedade Paranaense de Matemática, 2015

In this article, we study the following (p(x),q(x))(p(x),q(x))(p(x),q(x))-biharmonic type system \begin{gather*} \Del... more In this article, we study the following (p(x),q(x))(p(x),q(x))(p(x),q(x))-biharmonic type system \begin{gather*} \Delta(|\Delta u|^{p(x)-2}\Delta u)=\lambda F_u(x,u,v)\quad\text{in }\Omega,\\ \Delta(|\Delta v|^{q(x)-2}\Delta v)=\lambda F_v(x,u,v)\quad\text{in }\Omega,\\ u=v=\Delta u=\Delta v=0\quad \text{on }\partial\Omega. \end{gather*} We prove the existence of infinitely many solutions of the problem byapplying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

Electronic Journal of Qualitative Theory of Differential Equations, 2014

In this article, we study the nonlinear Steklov boundary-value problem ∆ p(x) u = |u| p(x)−2 u in... more In this article, we study the nonlinear Steklov boundary-value problem ∆ p(x) u = |u| p(x)−2 u in Ω, |∇u| p(x)−2 ∂u ∂ν = f (x, u) on ∂Ω. We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biha... more In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biharmonic ∆ 2 p u = λm(x)|u| p−2 u in a smooth bounded domain under Neumann boundary conditions.

In this note, we establish a variant of Ekeland's variational principle. This result suggests... more In this note, we establish a variant of Ekeland's variational principle. This result suggests a generalization of the classical Palais-Smale condition. An example is provided showing how this is used to give the existence of a minimizer for functionals which do not satisfy the Palais-Smale condition and the one introduced by Cerami. We also prove a relation between the coercitivity of functional and the introduced compactness condition.