An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systemsü(t) − L(t)u(t) + ∇W (t, u(t)) = 0, ∀t ∈ R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t ∈ R, and W (t, x) satisfies the superquadratic condition W (t, x)/|x| 2 → +∞ as |x| → ∞ uniformly in t, and need not satisfy the global AmbrosettiRabinowitz condition.

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

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