Some critical point results for Fr\'{e}chet manifolds

TL;DR

This paper establishes critical point theorems for functionals on Fréchet manifolds and provides conditions under which local diffeomorphisms are global, advancing the understanding of infinite-dimensional analysis.

Contribution

It introduces a linking theorem, mountain pass theorem, and three critical points theorem for Keller $C^1$-functionals on Fréchet manifolds, using a novel deformation approach.

Findings

Proved a linking theorem for Fréchet manifolds.

Derived a mountain pass theorem and a three critical points theorem.

Provided sufficient conditions for local diffeomorphisms to be global.

Abstract

We prove a so-called linking theorem and some of its corollaries, namely a mountain pass theorem and a three critical points theorem for Keller $ C^1$-functional on $ C^1 $- Frechet manifolds. Our approach relies on a deformation result which is not implemented by considering the negative pseudo-gradient flows. Furthermore, for mappings between Frechet manifolds we provide a set of sufficient conditions in terms of the Palais-Smale condition that indicates when a local diffeomorphism is a global one.