  We develop the local Morse theory for a class of non-twice continuously
differentiable functionals on Hilbert spaces, including a new generalization of
the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation
type result. With them some critical point theorems and famous bifurcation
theorems are generalized. Then we show that these are applicable to studies of
quasi-linear elliptic equations and systems of higher order given by
multi-dimensional variational problems as in (1.3).
