In the present section, we shall develop methods, employing ideas contained in some of L.A. Lyusternik's work, which allow us to establish the existence of a denumerable number of stable critical values of an even functional – they do not disappear under small perturbations by odd functionals.

This chapter is devoted to the study of the symmetric MPT and its subsequent extensions. It is a multiplicity result asserting the existence of multiple critical points, when the functional is invariant under the action of a group of symmetries. It has been stated in the same time as the classical MPT by Ambrosetti and Rabinowitz [50]. This theorem can be seen as an extension of older multiplicity results of Ljusternik Schnirelman type. We will also review two other ways of obtaining multiplicity results; a procedure that inductively uses the (classical) MPT and does not pass by any Index theory, and a generalization of the symmetric MPT, the fountain theorem of Bartsch and its dual form by Bartsch and Willem.