no. 5
Partial Differential Equations
A mountain pass theorem without Palais–Smale condition
[Un théorème du col sans condition de Palais–Smale]
Présenté par : Haïm Brezis
Lucia, Marcello1
1 Department of Mathematics, Rutgers University, Hill-Center, Piscataway, NJ 08854, USA
Comptes Rendus. Mathématique, Tome 341 (2005) no. 5, pp. 287-291

Given a Hilbert space (H,〈⋅,⋅〉), Λ an interval of R and J∈C2(H,R) whose gradient ∇J:H→H is a compact mapping, we consider a family of functionals of the type:

I(λ,u)=〈u,u〉−λJ(u),(λ,u)∈Λ×H.
Without further compactness assumptions, we present a deformation lemma to detect critical points. In particular, if I(λ¯,⋅) has a ‘mountain pass structure’ for some λ¯∈Λ, we deduce the existence of a sequence λn→λ¯ for which each I(λn,⋅) has a critical point.

Étant donné un espace de Hilbert (H,〈⋅,⋅〉), Λ un intervalle de R et J∈C2(H,R) dont le gradient ∇J:H→H est une application compacte, nous considérons une famille de fonctionelle de la forme :

I(λ,u)=〈u,u〉−λJ(u),(λ,u)∈Λ×H.
Sans autres hypothèses de compacité, nous présentons un lemme de déformation pour détecter des points critiques. En particulier, si I(λ¯,⋅) a une structure de « col » pour un certain λ¯∈Λ, nous montrons l'existence d'une suite λn→λ¯ pour laquelle chaque I(λn,⋅) admet un point critique.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.07.022

Lucia, Marcello 1

1 Department of Mathematics, Rutgers University, Hill-Center, Piscataway, NJ 08854, USA 
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Lucia, Marcello. A mountain pass theorem without Palais–Smale condition. Comptes Rendus. Mathématique, Tome 341 (2005) no. 5, pp. 287-291. doi: 10.1016/j.crma.2005.07.022

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