A further three critical points theorem

If X is a real Banach space, we denote by W(x) the class of all functionals phi : X -> R possessing the following property: if {u(n)} is a sequence in X converging weakly to u is an element of X and lim inf(n ->infinity) phi(u(n)) <= phi(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 subset of R an interval; phi : 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 lambda is an element of I, the functional phi - lambda J is coercive and has a strict local, not global minimum, say (x) over cap (lambda). Then, for each compact interval [a, b] subset of I for which sup(lambda is an element of[a,b]) (phi((X) over cap (lambda)) - lambda J((X) over cap (lambda))) < + infinity exists r > 0 with the following property: for every lambda is an element of [a, b] and every C(1) functional psi : X -> R with compact derivative, there exists delta > 0 such that, for each mu is an element of [0, delta], the equation phi'(x) = lambda J'(x) + mu psi'(x) has at least three solutions whose norms are less than r. (C) 2009 Elsevier Ltd. All rights reserved.