In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar ... more In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres S 3 , S 4 . Under generic conditions we establish some Morse Inequalities at Infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments we derive a new existence result on S 4 through an Euler-Hopf type formula. Mathematics Subject Classification (2000): 58E05 (primary); 35J65, 53C21, 35B40 (secondary).
This paper is devoted to the problem of prescribing the scalar curvature under zero boundary cond... more 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.
Let (S", c) be an n-dimensional sphere with a standard metric. Let K be a C2-positive fu... more Let (S", c) be an n-dimensional sphere with a standard metric. Let K be a C2-positive function on Sn. The Kazdan-Warner problem [12] is theproblem of finding suitable conditions on K such that K is thescalar-curvature for a metric O on Sn conformally equivalent to c. ...
... ON THE PRESCRIBED SCALAR CURVATURE PROBLEM ON 4-MANIFOLDS MOHAMED BENAYED, YANSONG CHEN, HICH... more ... ON THE PRESCRIBED SCALAR CURVATURE PROBLEM ON 4-MANIFOLDS MOHAMED BENAYED, YANSONG CHEN, HICHEM CHTIOUI, AND MOKHLES HAMMAMI 1. Introduction. ... 633 Page 2. 634 BEN AYED, CHEN, CHTIOUI, AND HAMMAMI construct. ...
... Author(s): M. BEN AYED Département de Mathématiques, Faculté des Sciences de Sfax, Route Souk... more ... Author(s): M. BEN AYED Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia K. EL MEHDI Centre de Mathématiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France O. REY Centre de Mathématiques, Ecole Polytechnique ...
This paper is concerned with a biharmonic equation under the Navier boundary condition with nearl... more 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 .
Nonlinear Differential Equations and Applications NoDEA, 2012
ABSTRACT In this paper, we consider the problem of prescribing the scalar curvature under minimal... more ABSTRACT In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.
Calculus of Variations and Partial Differential Equations, 2014
ABSTRACT We consider the problem of finding a positive harmonic function $u_\varepsilon $ in a bo... more ABSTRACT We consider the problem of finding a positive harmonic function $u_\varepsilon $ in a bounded domain $\Omega \subset \mathbb R ^N (N\ge 3)$ satisfying a nonlinear boundary condition of the form $\varepsilon \partial _{\nu } u +u =|u|^{p-2}u,\,x\in \partial \Omega $ , where $\varepsilon $ is a positive parameter and $2<p<2_*:=2(N-1)/(N-2)$ . To be more precise, by using min-max methods, we study the existence of least energy solution $u_\varepsilon $ of the problem depending on the parameter $\varepsilon $ . We provide a detailed description of the shape of $u_\varepsilon $ and prove that the maximum of $u_\varepsilon $ is achieved at a point $z_{\varepsilon }$ , which lies on the boundary $\partial \Omega $ and concentrates at the mean curvature maximum point of the boundary $\partial \Omega $ . This problem is related to the existence of extremals for a Sobolev inequality involving the trace embedding and the asymptotic behavior of the best constants in expanding domains.
In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar ... more In this paper, we consider the problem of multiplicity of conformal metrics of prescribed scalar curvature on standard spheres S 3 , S 4 . Under generic conditions we establish some Morse Inequalities at Infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler-Lagrange functional. As a by-product of our arguments we derive a new existence result on S 4 through an Euler-Hopf type formula. Mathematics Subject Classification (2000): 58E05 (primary); 35J65, 53C21, 35B40 (secondary).
This paper is devoted to the problem of prescribing the scalar curvature under zero boundary cond... more 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.
Let (S", c) be an n-dimensional sphere with a standard metric. Let K be a C2-positive fu... more Let (S", c) be an n-dimensional sphere with a standard metric. Let K be a C2-positive function on Sn. The Kazdan-Warner problem [12] is theproblem of finding suitable conditions on K such that K is thescalar-curvature for a metric O on Sn conformally equivalent to c. ...
... ON THE PRESCRIBED SCALAR CURVATURE PROBLEM ON 4-MANIFOLDS MOHAMED BENAYED, YANSONG CHEN, HICH... more ... ON THE PRESCRIBED SCALAR CURVATURE PROBLEM ON 4-MANIFOLDS MOHAMED BENAYED, YANSONG CHEN, HICHEM CHTIOUI, AND MOKHLES HAMMAMI 1. Introduction. ... 633 Page 2. 634 BEN AYED, CHEN, CHTIOUI, AND HAMMAMI construct. ...
... Author(s): M. BEN AYED Département de Mathématiques, Faculté des Sciences de Sfax, Route Souk... more ... Author(s): M. BEN AYED Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia K. EL MEHDI Centre de Mathématiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France O. REY Centre de Mathématiques, Ecole Polytechnique ...
This paper is concerned with a biharmonic equation under the Navier boundary condition with nearl... more 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 .
Nonlinear Differential Equations and Applications NoDEA, 2012
ABSTRACT In this paper, we consider the problem of prescribing the scalar curvature under minimal... more ABSTRACT In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.
Calculus of Variations and Partial Differential Equations, 2014
ABSTRACT We consider the problem of finding a positive harmonic function $u_\varepsilon $ in a bo... more ABSTRACT We consider the problem of finding a positive harmonic function $u_\varepsilon $ in a bounded domain $\Omega \subset \mathbb R ^N (N\ge 3)$ satisfying a nonlinear boundary condition of the form $\varepsilon \partial _{\nu } u +u =|u|^{p-2}u,\,x\in \partial \Omega $ , where $\varepsilon $ is a positive parameter and $2<p<2_*:=2(N-1)/(N-2)$ . To be more precise, by using min-max methods, we study the existence of least energy solution $u_\varepsilon $ of the problem depending on the parameter $\varepsilon $ . We provide a detailed description of the shape of $u_\varepsilon $ and prove that the maximum of $u_\varepsilon $ is achieved at a point $z_{\varepsilon }$ , which lies on the boundary $\partial \Omega $ and concentrates at the mean curvature maximum point of the boundary $\partial \Omega $ . This problem is related to the existence of extremals for a Sobolev inequality involving the trace embedding and the asymptotic behavior of the best constants in expanding domains.
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