On locally superquadratic Hamiltonian systems with periodic potential

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