23 - Perturbation from Symmetry and the MPT

Summary

In this chapter, we are interested in some methods used to find multiplicity results, as in the symmetric MPT, when the symmetry is broken by a “little” perturbation added to the functional. We focus on two methods closely related to the MPT.

As we saw in Chapter 11, when the functional under study is equivariant, corresponding MPTs yield multiplicity results. These have important applications in partial differential equations and Hamiltonian systems that are invariant under a group action.

A natural question that stems then is whether it is really the symmetry that is responsible for these spectacular multiplicity results. Would the results persist if some perturbation is injected? This stability problem is quite old, and mathematicians were interested in the study of the effects of breaking the symmetry by introducing a small perturbation since the appearance of the Ljusternik-Schnirelman theory.

… we shall develop methods, employing ideas contained in some of L.A. Ljusternik'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.

Krasnosel'skii [534]