On locally superquadratic Hamiltonian systems with periodic potential

In this paper, we study the second-order Hamiltonian systems u ¨ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L and W depend periodically on t , 0 lies in a spectral gap of the operator − d 2 / d t 2 + L ( t ) and W ( t , x ) is locally superquadratic. Replacing the common superquadratic condition that lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ uniformly in t ∈ R by the local condition that lim | x | → ∞ W ( t , x ) | x | 2 = + ∞ a.e. t ∈ J for some open interval J ⊂ R , we prove the existence of one nontrivial homoclinic soluiton for the above problem.