Homoclinic solutions for some second order non-autonomous Hamiltonian systems without the globally superquadratic condition

摘要

This paper deals with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system q + V_q(t,q)=f(t), (HS) where V ∈C~1(R × R~n, R), V(t,q) =-K(t,q)+W(t,q) is T -periodic in t, f is aperiodic and belongs to L2.R; Rn/. Under the assumptions that K satisfies the "pinching" condition b_1｜q｜~2 ≤ K(t,q) ≤ b_2｜q｜~2, W(t,q) is not globally superquadratic on q and some additionally reasonable assumptions, we give a new existence result to guarantee that (HS) has a homoclinic solution q(t) emanating from 0. The homoclinic solution q(t) is obtained as a limit of 2kT -periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.