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

Abstract

This paper deals with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system (HS) q ̈ + V q ( t , q ) = f ( t ) , 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 L 2 ( R , R n ) . 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 2 k T -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.