Infinitely homoclinic solutions in discrete hamiltonian systems without coercive conditions

Abstract

In this paper, we investigate the existence of infinitely many solutions for the second-order self-adjoint discrete Hamiltonian system
$$
\Delta\left[p(n)\Delta u(n-1)\right]-L(n)u(n)+\nabla W(n,u(n))=0, \tag{*}
$$
where \(n\in\mathbb{Z}, u\in\mathbb{R}^{N}, p,L:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}\) and \(W:\mathbb{Z}\times\mathbb{R}^{N}\rightarrow\mathbb{R}\) are no periodic in \(n\). The novelty of this paper is that \(L(n)\) is bounded in the sense that there two constants \(0<\tau_1<\tau_2<\infty\) such that
$$
\tau_1\left|u\right|^{2}<\left(L(n)u,u\right)<\tau_2\left|u\right|^{2},\;\forall n\in\mathbb{Z},\; u\in\mathbb{R}^{N},
$$
\(W(t,u)\) satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that (\(*\)) has infinitely many homoclinic solutions via the Symmetric Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.