Homoclinic solutions for p(t)-Laplacian Hamiltonian systems with new conditions

The existence of homoclinic solutions is obtained for a class of p(t)-Laplacian Hamiltonian systems d dt (| u(t)| p(t)-2 u(t))a(t)|u(t)| p(t)-2 u(t) + W (t, u(t)) = 0 via variational methods, where a(t) is neither coercive nor bounded necessarily, and W (t, u) is under new super-p(t) growth conditions.