A three critical points theorem revisited

Abstract

In this paper the following result is proved: Let X be a reflexive real Banach space; I ⊆ R an interval; Φ : X → R a sequentially weakly lower semicontinuous C 1 functional, bounded on each bounded subset of X , whose derivative admits a continuous inverse on X ∗ ; J : X → R a C 1 functional with compact derivative. Assume that lim ‖ x ‖ → + ∞ ( Φ ( x ) + λ J ( x ) ) = + ∞ for all λ ∈ I , and that there exists ρ ∈ R such that sup λ ∈ I inf x ∈ X ( Φ ( x ) + λ ( J ( x ) + ρ ) ) < inf x ∈ X sup λ ∈ I ( Φ ( x ) + λ ( J ( x ) + ρ ) ) . Then, there exist a non-empty open set A ⊆ I and a positive real number r with the following property: for every λ ∈ A and every C 1 functional Ψ : X → R with compact derivative, there exists δ > 0 such that, for each μ ∈ [ 0 , δ ] , the equation Φ ′ ( x ) + λ J ′ ( x ) + μ Ψ ′ ( x ) = 0 has at least three solutions in X whose norms are less than r .