The splitting lemmas for nonsmooth functionals on Hilbert spaces I

Abstract

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)={\int }_{\Omega }f(x,u,...,D^{m}u)dx$ as in (\ref{e:1.1}). 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 Poincaré-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the 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 systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.

68 pages; v2: for the published version we correct a few typo and state Theorems A.1, A.2 in a more precise way

References in corpus (2)

• Some critical point theorems and applications
• Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds

Cited by in corpus (4)

• Some critical point theorems and applications
• Splitting lemmas for the Finsler energy functional on the space of $H^1$-curves
• Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds
• The splitting lemmas for nonsmooth functionals on Hilbert spaces II. The case at infinity