A MOUNTAIN PASS THEOREM FOR A SUITABLE CLASS OF FUNCTIONS DIEGO AVERNA AND GABRIELE BONANNO Abstract. The main purpose of this paper is to establish a three critical points result without assuming the coercivity of the involved functional. To this end, a mountain-pass theorem, where the usual Palais-Smale condition is not requested, is presented. These results are then applied to prove the existence of three solutions for a two-point boundary value problem with no asymptotic conditions. 1. Introduction It is well known that the mountain-pass theorem of A. Ambrosetti and P.H. Rabi- nowitz [1, Theorem 2.1] and its variants or generalizations, as for instance Theorem 1 of [17], is successfully used to find critical points of real-valued C 1 functions J defined on an infinite-dimensional Banach space X. One of the key assumptions in this result is a compactness hypothesis, usually called Palais-Smale condition. The present paper deals with the case J (x) = Φ(x) − Ψ(x), x ∈ X, which often occurs in the variational formulation of both ordinary and partial differ- ential problems. We first introduce a new type of Palais-Smale condition; see Section 3. It is mutually independent from the usual condition and holds true every time that Φ, Ψ turn out sufficiently smooth and Φ is coercive; see Theorem 3.1. A moun- tain pass-like result, which also provides a more precise localization of the obtained critical point as regards the function Φ, is then established (see Theorem 4.3). More- over, putting this result and Theorem 2.1 in [5] together yields the main result of the paper, which is a three critical points theorem for the functional J λ (x) = Φ(x) − λΨ(x), x ∈ X, where λ> 0, whose norms are uniformly bounded with respect to λ; see Theorem 5.2. Let us point out that, contrary to the basic result of B. Ricceri [20, Theorem 1] on this topic, no coercivity of J λ is assumed. To prove our results we employ tools from Key words and phrases. Palais-Smale condition, mountain pass, critical points, three solutions, two-point boundary value problem. 2000 Mathematics Subject Classification: 47J30, 58E30, 49J40, 58E05, 34B15. Corresponding author: G.Bonanno. This research was partially supported by RdB (ex 60% MIUR) of Reggio Calabria University. Typeset by L A T E X2 ε . 1