Connecting orbits for some classes of almost periodic Lagrangian systems
For a class of almost-periodic Lagrangian systems the authors prove, by means of variational arguments, the existence of infinitely many heteroclinic solutions connecting possibly degenerate equilibria. The systems dealt with are of the form \[ \ddot q+\alpha (t)\bigtriangledown V(q)=0, \eqno (1) \] where \((\text{ H}_1)\) \(\alpha\) is a continuous almost-periodic function with \(\alpha (t)\geq \underline {\alpha}>0\) for all \(t\in {\mathbb{R}}\), \((\text{ H}_2)\) \(V\in C^1({\mathbb{R}}^N)\) and \(V(x+\zeta)=V(x)\) for all \(x\in {\mathbb{R}}^N\), \(\zeta\in {\mathbb{Z}}^N\), \((\text{ H}_3)\) \(V(x)< V(\zeta)=0\) for any \(\zeta\in {\mathbb{Z}}^N\), \(x\in {\mathbb{R}}^N\setminus {\mathbb{Z}}^N\). Assumptions \((\text{ H}_1) - (\text{ H}_3)\) yield that any \(\zeta\in {\mathbb{Z}}^N\) is an equilibrium and heteroclinic solutions between these equilibria are the subject of studies. The main result is the following. Theorem. If \((\text{ H}_1) - (\text{ H}_3)\) hold, then system (1) admits infinitely many heteroclinic solutions. More precisely, there exist \(K\in {\mathbb{N}}\), \(K\geq 2N\), and \(K\) points \(\zeta _1, \dots ,\zeta _K \in {\mathbb{Z}}^N\setminus \{0\}\) such that \(\{\sum_{i=1}^{K}n_i\zeta _i : n_i\in {\mathbb{N}}\cup \{0\}\}={\mathbb{Z}}^N\) and such that for any fixed \(j\in\{ 1,\dots ,K\}\) there exist infinitely many heteroclinic solutions to (1) connecting \(0\) with \(\zeta _j\). An analogous multiplicity result is established for homoclinic solutions to systems with an almost-periodic singular potential.
scientific article; zbMATH DE number 1206628