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Heteroclinic solutions for a class of the second order Hamiltonian systems. (English) Zbl 1117.37033

Summary: We are concerned with the existence of heteroclinic orbits for the second-order Hamiltonian system \(\ddot q+V_q(t,q)=0\), where \(q\in \mathbb{R}^n\) and \(V\in C^1(\mathbb{R} \times\mathbb{R}^n,\mathbb{R})\), \(V\leq 0\). We will assume that \(V\) and a certain subset \({\mathcal M}\subset\mathbb{R}^n\) satisfy the following conditions. \({\mathcal M}\) is a set of isolated points and \(\# {\mathcal M}\geq 2\). For every sufficiently small \(\varepsilon >0\) there exists \(\delta>0\) such that for all \((t,z)\in\mathbb{R}\times\mathbb{R}^n\), if \(d(z,{\mathcal M}) \geq\varepsilon\) then \(-V(t,z)\geq\delta\). The integrals \(\int^\infty_{-\infty}-V(t,z)dt\), \(z\in{\mathcal M}\), are equibounded and \(-V(t,z) \to\infty\), as \(|t|\to\infty\), uniformly on compact subsets of \(\mathbb{R}^n\setminus {\mathcal M}\). Our result states that each point in \({\mathcal M}\) is joined to another point in \({\mathcal M}\) by a solution of our system.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
70H05 Hamilton’s equations
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