A further three critical points theorem

If X is a real Banach space, we denote by W X the class of all functionals Φ : X → R possessing the following property: if { u n } is a sequence in X converging weakly to u ∈ X and lim inf n → ∞ Φ ( u n ) ≤ Φ ( u ) , then { u n } has a subsequence converging strongly to u . In this paper, we prove the following result: Let X be a separable and reflexive real Banach space; I ⊆ R an interval; Φ : X → R a sequentially weakly lower semicontinuous C 1 functional, belonging to W X , bounded on each bounded subset of X and whose derivative admits a continuous inverse on X ∗ ; J : X → R a C 1 functional with compact derivative. Assume that, for each λ ∈ I , the functional Φ − λ J is coercive and has a strict local, not global minimum, say x ˆ λ . Then, for each compact interval [ a , b ] ⊆ I for which sup λ ∈ [ a , b ] ( Φ ( x ˆ λ ) − λ J ( x ˆ λ ) ) < + ∞ , there exists r > 0 with the following property: for every λ ∈ [ a , b ] 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 ) has at least three solutions whose norms are less than r .