A Generalized Palais-Smale Condition in the Fr\'echet space setting

TL;DR

This paper extends the Palais-Smale condition to Fréchet spaces and demonstrates its implications for the existence of minima using variational principles, broadening the functional analysis framework.

Contribution

It introduces a generalized Palais-Smale condition for Keller's $C_c^1$-functionals on Fréchet spaces, linking it to coercivity and minima existence.

Findings

Palais-Smale condition extended to Fréchet spaces

Existence of minima established via variational principles

Palais-Smale condition implies coercivity for bounded below functionals

Abstract

We extend the Palais-Smale condition to Keller's $C_c^1$-functionals on Fr\'{e}chet spaces. Using this condition together with Ekeland's variational principle, we obtain some results regarding the existence of minima. In this setting, we prove that the Palais-Smale condition for functionals bounded below implies the coercivity.