The splitting lemmas for nonsmooth functionals on Hilbert spaces II. The case at infinity

TL;DR

This paper extends the splitting lemma at infinity for a broader class of functionals on Hilbert spaces, combining Morse-Palais techniques with new methods, and provides an application.

Contribution

It generalizes the splitting lemma at infinity to continuously directional differentiable functionals using a novel proof approach.

Findings

Extended the splitting lemma at infinity to new functional classes

Combined Morse-Palais ideas with flow method techniques

Presented a simple application of the generalized lemma

Abstract

We generalize the Bartsch-Li's splitting lemma at infinity for $C^2$-functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow methods our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [9] with some techniques from [11], [17], [18]. A simple application is also presented.