This paper establishes a new three critical points theorem for the equation
under specific hypotheses. If is real Banach space, denote by the class of functionals possessing the following property: if is a sequence in converging weakly to and , then has a subsequence converging strongly to .
The main result of the paper is as follows.
Theorem 1. Let be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous functional from bounded on each bounded subset of and whose derivative admits a continuous inverse on ; a functional with compact derivative. Assume that, for each , the functional is coercive and has a strict local, not global minimum, say .
Then, for each compact interval for which , there exists with the following property: for every and every functional with compact derivative, there exists such that, for each , the equation
has at least three solutions whose norms are less than .
Some applications of this result are also given.