Two Almost Homoclinic Solutions for a Class of Perturbed Hamiltonian Systems Without Coercive Conditions

Abstract

In this paper we consider the existence of almost homoclinic solutions for the following second order perturbed Hamiltonian systems  ü- L(t)u + ∇W (t, u) = f (t), (PHS)  where  is a symmetric and positive definite matrix for all t ∈ R, W ∈ C¹(R×Rⁿ, R) and ∇W (t, u) is the gradient of W (t, u) at u, f ∈ C(R, Rⁿ) and belongs to L²(R, Rⁿ). The novelty of this paper is that, assuming L(t) is bounded in the sense that there are two constants 0 < τ1 < τ2 <  ∞ such that τ1 ∣u∣² ≤ (L(t)u, u) ≤ τ2 ∣u∣² for all (t, u) ∈ R × Rⁿ, W(t, u) satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, f (t) is sufficiently small in L²(R, Rⁿ), we obtain some new criterion to guarantee that (PHS) has at least two nontrivial almost homoclinic solutions. Recent results in the literature are generalized and significantly improved.

Keyword : homoclinic solutions, critical point, variational methods, mountain pass theorem