A mountain pass theorem without Palais–Smale condition

Abstract

Given a Hilbert space ( H , 〈 ⋅ , ⋅ 〉 ) , Λ an interval of R and J ∈ C 2 ( 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 〉 − λ J ( 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 λ n → λ ¯ for which each I ( λ n , ⋅ ) has a critical point. To cite this article: M. Lucia, C. R. Acad. Sci. Paris, Ser. I 341 (2005).