The splitting lemmas for nonsmooth functionals on Hilbert spaces

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.