  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.
