The splitting lemmas for nonsmooth functionals on Hilbert spaces

TL;DR

This paper extends splitting lemmas to a broader class of nonsmooth functionals on Hilbert spaces, enabling analysis of critical points with lower smoothness and applications to nonlinear elliptic equations.

Contribution

It establishes new splitting and shifting theorems for lower than $C^1$ functionals on Hilbert spaces, generalizing previous results and connecting critical groups across different spaces.

Findings

Generalized splitting theorem for lower smoothness functionals

New Poincaré-Hopf type theorem for critical points

Applications to nonlinear elliptic boundary value problems

Abstract

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincar\'e-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. The corresponding version at critical submanifolds is presented. We also generalize the Bartsch-Li's splitting lemma at infinity in \cite{BaLi} and some variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Our proof methods are to combine the proof ideas of the Morse-Palais lemma due to Duc-Hung-Khai \cite{DHK} with some techniques from \cite{JM, Skr, Va1}. Our theory is applicable to the Lagrangian system on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.