A characterization of the mountain pass geometry for functionals bounded from below

Abstract

In this paper it is proved that, when a regular functional is bounded from below, the mountain pass geometry and the existence of at least two distinct local minima are equivalent conditions. As a consequence, the classical mountain pass theorem, under the additional assumption of boundedness from below of the functional, ensures actually three distinct critical points. Moreover, as application, the existence of three solutions to Hamiltonian systems is established.