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Khalil El Mehdi
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ABSTRACT: In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent $(P_\epsilon): \Delta^2u=u^{9-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\R^5$ and $\epsilon >0$. We study the asymptotic behavior of solutions of $(P_\epsilon)$ which are minimizing for the Sobolev qutient as $\epsilon$ goes to zero. We show that such solutions concentrate around a point $x_0\in\Omega$ as $\epsilon\to 0$, moreover $x_0$ is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point $x_0$ of the Robin's function, there exist solutions concentrating around $x_0$ as $\epsilon$ goes to zero.
01/2005;
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ABSTRACT: In this paper we consider the following biharmonic equation with critical exponent $P_\epsilon$ : $\Delta^2 u= Ku^{(n+4)/(n-4)-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a domain in $R^n$, $n\geq 5$, $\epsilon$ is a small positive parameter and $K$ is smooth positive function. We construct solutions of $P_\epsilon$ which blow up and concentrate at strict local maximum of $K$ either at the boundary or in the interior of $\Omega$. We also construct solutions of $P_\epsilon$ concentrating at an interior strict local minimum of $K$. Finally, we prove a nonexistense result for the corresponding supercritical problem which is in sharp contrast with what happened for $P_\epsilon$.
09/2004;
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ABSTRACT: We consider a Yamabe type problem on a family $A_\epsilon$ of annulus shaped domains of $\R^3$ which becomes "thin" as $\epsilon$ goes to zero. We show that, for any given positive constant $C$, there exists $\epsilon_0$ such that for any $\epsilon < \epsilon_0$, the problem has no solution $u_\epsilon$ whose energy is less than $C$. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions $u_\epsilon$ when $\epsilon$ goes to zero.
09/2004;
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ABSTRACT: This paper is concerned with a biharmonic equation under the Navier boundary condition with nearly critical exponent. We study the asymptotic behavior os solutions which are minimizing for the Sobolev quatient. We show that such solutions concentrate around an interior point which is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point fo the Robin's function, the exist solutions concentrating around such a point. Finally, we prove that, in contrast with what happened in the subcritical equation, the supercritical problem has no solutions which concentrate around a point .
02/2004;
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ABSTRACT: In this paper, we studu a biharmonic equation under the Navier boundary condition on thin annuli. We show that when the annulus becomes thin, the equation has no solution whose energy is bounded.
12/2003;
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ABSTRACT: This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results. These methods were first introduced by A. Bahri.
07/2003;
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ABSTRACT: In this paper we consider a fourth order equation involving the critical Sobolev exponent on a bounded and smooth domain in $\R^6$. Using theory of critical points at infinity, we give some topological conditions on a given function defined on a domain to ensure some existence results.
07/2003;
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ABSTRACT: In this paper we prescribe a fourth order conformal invariant 9the Paneitz Curvature) on five and six spheres. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results.
06/2003;
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ABSTRACT: In this paper a fourth order equation involving critical growth is considered under Navier boundary condition. We give some topological conditions on a given function to ensure the existence of solutions. Our methods involve the study of the critical points at infinity and their contribution to the topology of the level sets of the associated Euler Lagrange functional
06/2003;
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ABSTRACT: In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the $n$-sphere, with $n\geq 5$. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existene of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.
06/2003;
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ABSTRACT: This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.
06/2002;
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ABSTRACT: In this paper we study low energy sign changing solutions of the critical exponent problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in R3 and λ is a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameter λ which are related to the existence of positive or of sign changing solutions.RésuméDans cet article, nous étudions les solutions changeant de signe et à énergie minimale du problème avec exposant critique dans Ω, u=0 sur ∂Ω, où Ω est un domaine borné et régulier de R3 et λ est un paramètre réel strictement positif. Nous faisons une analyse précise du ‘blow-up’ de ce type de solutions et nous prouvons des résultats de comparaisons pour certaines valeurs limites du paramètre λ qui sont liées à l'existence des solutions positives ou des solutions changeant de signe.
Annales de l'Institut Henri Poincare (C) Non Linear Analysis.
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ABSTRACT: In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S3 of C2. Our methods are based on the techniques related to the theory of critical points at infinity.RésuméDans ce papier, nous prouvons quelques résultats d'existence pour le problème de la courbure scalaire de Webster sur les variétés CR de dimension trois localement conformement CR equivalent à la sphere unité S3 de C2. Nos méthodes sont basées sur des techniques liées à la théorie des points critiques à l'infini.
Bulletin des Sciences Mathématiques.
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ABSTRACT: In this paper a fourth order equation involving critical growth is considered under the Navier boundary condition: Δ2u=Kup, u>0 in Ω, u=Δu=0 on ∂Ω, where K is a positive function, Ω is a bounded smooth domain in Rn, n⩾5 and p+1=2n/(n−4), is the critical Sobolev exponent. We give some topological conditions on K to ensure the existence of solution. Our methods involve the study of the critical points at infinity and their contribution to the topology of the level sets of the associated Euler–Lagrange functional.RésuméDans cet article, nous considérons une équation d'ordre quatre ayant une croissance critique avec conditions de Navier au bord : Δ2u=Kup, u>0 dans Ω, u=Δu=0 sur ∂Ω, où K est une fonction strictement positive, Ω est un domaine borné régulier de Rn, n⩾5 et p+1=2n/(n−4), est l'exposant critique de Sobolev. Nous donnons certaines conditions topologiques sur K pour assurer l'existence de solution. Notre approche est fondée sur l'étude des points critiques à l'infini et de leur contribution à la topologie des ensembles de niveau de la fonctionnelle d'Euler–Lagrange associée.
Journal de Mathématiques Pures et Appliquées.
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ABSTRACT: In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball.
Journal of Differential Equations.
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ABSTRACT: In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball.
Journal of Functional Analysis.
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