A three critical points theorem revisited

In this paper the following result is proved: Let XX be a reflexive real Banach space; I⊆R an interval; Φ:X→R a sequentially weakly lower semicontinuous C1C1 functional, bounded on each bounded subset of XX, whose derivative admits a continuous inverse on X∗X∗; J:X→R a C1C1 functional with compact derivative. Assume that lim‖x‖→+∞(Φ(x)+λJ(x))=+∞ for all λ∈Iλ∈I, and that there exists ρ∈R such that supλ∈Iinfx∈X(Φ(x)+λ(J(x)+ρ))0δ>0 such that, for each μ∈[0,δ]μ∈[0,δ], the equation Φ′(x)+λJ′(x)+μΨ′(x)=0Φ′(x)+λJ′(x)+μΨ′(x)=0 has at least three solutions in XX whose norms are less than rr.