The Existence of Heteroclinic Orbits for a Class of the Second-Order Hamiltonian System

Abstract

In this paper, using variational methods and critical point theory, we shall mainly be concerned with the study of the existence of heteroclinic solutions for the following second-order Hamiltonian systems \({\ddot{u} + V_{u}(t, u) = 0}\) , where \({V \in C^{1}(\mathbb{R} \times \mathbb{R}^{N}, \mathbb{R})}\) and \({V \leq 0, u \in \mathbb{R}^{N}}\) . Setting \({{\mathcal{W}} = \{\xi \in \mathbb{R}^{N} | V(t, \xi) = 0 \, {\rm for} \, {\rm all} \,t \in \mathbb{R}\}}\) , we shall assume that \({V, {\mathcal{W}}}\) and \({{\mathcal{N}} \subset \mathbb{R}}\) consisting only of isolated point satisfy the following conditions: \({\sharp{\mathcal{W}} \geq 2}\) and \({\gamma_{0} \doteq \frac{1}{3}{\rm inf}\{| \xi - \eta |: \xi, \eta \in {\mathcal{W}}, \xi \neq\eta \} > 0}\) . For every \({0 0 such that for all \({(t, x) \in \mathbb{R} \times \mathbb{R}^{N}}\) , if \({d(x, {\mathcal{W}}) \geq \varepsilon}\) , then −V(t, x) ≥ δ. For any sufficiently small \({\bar{\varepsilon} > 0}\) and \({x \in \mathbb{R}^{N}\backslash{\mathcal{W}}, -V(t, x) \rightarrow \infty}\) if \({t \not\in B_{\bar{\varepsilon}}(\mathcal{N})}\) and |t| → ∞. Under these assumptions, we prove that each point in \({\mathcal{W}}\) is joined to another point in \({\mathcal{W}}\) by a solution of our system.