A mountain pass theorem without Palais–Smale condition

Marcello Lucia

doi:10.1016/j.crma.2005.07.022

Partial Differential Equations

A mountain pass theorem without Palais–Smale condition

Presented by: Haïm Brezis

Marcello Lucia ¹

¹Department of Mathematics, Rutgers University, Hill-Center, Piscataway, NJ 08854, USA

Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 287-291

Abstract (Original language)
Résumé

Given a Hilbert space $(H, 〈 \cdot, \cdot 〉)$ , Λ an interval of $R$ and $J \in C^{2} (H, R)$ whose gradient $\nabla J : H \to H$ is a compact mapping, we consider a family of functionals of the type:

I (λ, u) = 〈 u, u 〉 - λ J (u), (λ, u) \in Λ \times H .

Without further compactness assumptions, we present a deformation lemma to detect critical points. In particular, if

I (\bar{λ}, \cdot)

has a ‘mountain pass structure’ for some

\bar{λ} \in Λ

, we deduce the existence of a sequence

λ_{n} \to \bar{λ}

for which each

I (λ_{n}, \cdot)

has a critical point.

Étant donné un espace de Hilbert $(H, 〈 \cdot, \cdot 〉)$ , Λ un intervalle de $R$ et $J \in C^{2} (H, R)$ dont le gradient $\nabla J : H \to H$ est une application compacte, nous considérons une famille de fonctionelle de la forme :

I (λ, u) = 〈 u, u 〉 - λ J (u), (λ, u) \in Λ \times H .

Sans autres hypothèses de compacité, nous présentons un lemme de déformation pour détecter des points critiques. En particulier, si

I (\bar{λ}, \cdot)

a une structure de « col » pour un certain

\bar{λ} \in Λ

, nous montrons l'existence d'une suite

λ_{n} \to \bar{λ}

pour laquelle chaque

I (λ_{n}, \cdot)

admet un point critique.

Accepted: 2005-07-17
Published online: 2005-08-30

DOI: 10.1016/j.crma.2005.07.022

Author's affiliations:

Marcello Lucia ¹

¹ Department of Mathematics, Rutgers University, Hill-Center, Piscataway, NJ 08854, USA

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     author = {Marcello Lucia},
     title = {A mountain pass theorem without {Palais{\textendash}Smale} condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {287--291},
     year = {2005},
     publisher = {Elsevier},
     volume = {341},
     number = {5},
     doi = {10.1016/j.crma.2005.07.022},
     language = {en},
}

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Marcello Lucia. A mountain pass theorem without Palais–Smale condition. Comptes Rendus. Mathématique, Volume 341 (2005) no. 5, pp. 287-291. doi: 10.1016/j.crma.2005.07.022

References
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[3] W. Ding; J. Jost; J. Li; G. Wang Existence results for mean field equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 16 (1999), pp. 653-666

[4] M. Struwe; G. Tarantello On multivortex solutions in Chern–Simons Gauge theory, Boll. U.M.I. B (8), Volume 1 (1998), pp. 109-121

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