Homoclinic solutions for second order Hamiltonian systems with small forcing terms

Abstract

The existence of homoclinic solutions is obtained for a class of nonautonomous second order Hamiltonian systems $\ddot{u}(t)+\nabla V(t,u(t))=f(t)$ as the limit of the $2kT$-periodic solutions which are obtained by the Mountain Pass theorem, where $V(t,x)=-K(t,x)+W(t,x)$ is $T$-periodic with respect to $t,T>0$, and $W(t,x)$ satisfies the superquadratic condition: $W(t,x) / |x|^{2} \rightarrow +\infty$ as $|x| \rightarrow \infty$ uniformly in $t$, which needs not to satisfy the global Ambrosetti-Rabinowitz condition.