Homoclinic solutions for second order Hamiltonian systems with general potentials

Abstract

AbstractIn this paper we are concerned with the existence of infinitely many homoclinic solutions for the following second order non-autonomous Hamiltonian systemsu¨t−Ltut+∇Wt,ut=0$$ \ddot u\left( t \right) - L\left( t \right)u\left( t \right) + \nabla W\left( {t,u\left( t \right)} \right) = 0$$(HS)wheret∈ ℝ,L∈C(ℝ, ℝn2) is a symmetric and positive definite matrix for allt∈ ℝ,W∈C1(ℝ × ℝn, ℝ) and ∇W(t,u) is the gradient ofWatu. The novelty of this paper is that, assuming thatLmeets some coercive condition and the potentialWis of the formW(t,u) =W1(t,u) +W2(t,u), for the first time we show that (HS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (HS) goes to infinity and zero, respectively. Some recent results in the literature are generalized and significantly improved.