16 - The Metric MPT

Summary

Mathematical discoveries, small or great are never born of spontaneous generation. They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.

Bertrand Russel

Going further in nonsmoothness, we present now a variant of the MPT for continuous functionals on metric spaces. Appropriate notions of critical point and Palais-Smale condition are defined to handle this more general situation that still contains as particular cases the previous results stated when more smoothness and regularity on the functional were supposed.

The metric MPT was discovered independently by Degiovanni and Marzocchi [310] and Katriel [516]. They both use Ekeland's variational principle but in two different ways. The method of Degiovanni and Marzocchi has become widely known these days. Nevertheless, Katriel's approach will be more familiar to those who have read the previous chapter devoted to the nonsmooth MPT.

Preliminaries

In both papers [310, 516], the notions of critical point and Palais-Smale condition are defined in a very similar way although they use different terminology.

Critical Points of Continuous Functions in Metric Spaces

The definition of a critical point for a continuous function defined on a metric space reduces to the usual one known in the smooth case when the functional is smooth.