A three critical points theorem and its applications to the ordinary Dirichlet problem

Abstract

The aim of this paper is twofold. On one hand we establish a three critical points theorem for functionals depending on a real parameter $\lambda \in \Lambda$, which is different from the one proved by B. Ricceri in [On a three critical points theorem, Arch. Math. 75 (2000), 220-226] and gives an estimate of where $\Lambda$ can be located. On the other hand, as an application of the previous result, we prove an existence theorem of three classical solutions for a two-point boundary value problem which is independent from the one by J. Henderson and H. B. Thompson [Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166 (2000), 443-454]. Specifically, an example is given where the key assumption of [J. Differential Equations 166 (2000), 443-454] fails. Nevertheless, the existence of three solutions can still be deduced using our theorem.