A further three critical points theorem

If XX is a real Banach space, we denote by WXWX the class of all functionals Φ:X→R possessing the following property: if {un}{un} is a sequence in XX converging weakly to u∈Xu∈X and lim infn→∞Φ(un)≤Φ(u)lim infn→∞Φ(un)≤Φ(u), then {un}{un} has a subsequence converging strongly to uu.In this paper, we prove the following result:Let XX be a separable and reflexive real Banach space; I⊆R an interval; Φ:X→R a sequentially weakly lower semicontinuous C1C1 functional, belonging to WXWX, bounded on each bounded subset of XX and whose derivative admits a continuous inverse on X∗X∗; J:X→R a C1C1 functional with compact derivative. Assume that, for each λ∈Iλ∈I, the functional Φ−λJΦ−λJ is coercive and has a strict local, not global minimum, say xˆλ.Then, for each compact interval [a,b]⊆I[a,b]⊆I for which supλ∈[a,b](Φ(xˆλ)−λJ(xˆλ))<+∞, there exists r>0r>0 with the following property: for every λ∈[a,b]λ∈[a,b] and every C1C1 functional Ψ:X→R with compact derivative, there exists δ>0δ>0 such that, for each μ∈[0,δ]μ∈[0,δ], the equation Φ′(x)=λJ′(x)+μΨ′(x)Φ′(x)=λJ′(x)+μΨ′(x) has at least three solutions whose norms are less than rr.