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Prescribing Q -curvature on higher dimensional spheres

Khalil El Mehdi — 2005

Annales mathématiques Blaise Pascal

We study the problem of prescribing a fourth order conformal invariant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.

Morse index and blow-up points of solutions of some nonlinear problems

Khalil El MehdiFilomena Pacella — 2002

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we consider the following problem - u = N N - 2 u p ϵ - λ u in  Ω u > 0 in  Ω u = 0 on  Ω . where Ω is a bounded smooth starshaped domain in R N , N 3 , p ϵ = N + 2 N - 2 - ϵ , ϵ > 0 , and λ 0 . We prove that if u ϵ is a solution of Morse index m > 0 than u ϵ cannot have more than m maximum points in Ω for ϵ sufficiently small. Moreover if Ω is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for ϵ sufficiently small.

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben AyedKhalil El MehdiMohameden Ould AhmedouFilomena Pacella — 2005

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : Δ u ε = u ε ( n + 2 ) / ( n 2 ) , u ε > 0 in A ε , u ε = 0 on A ε , where A ε is an annulus-shaped domain of n , n 3 , which becomes thinner as ε 0 . We show that for every solution u ε , the energy A ε | u | 2 as well as the Morse index tend to infinity as ε 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

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