Multiple homoclinic orbits for autonomous, singular potentials. (English) Zbl 0812.58088

This work shows that the boundary value problem \[ \left\{ \begin{aligned} \ddot u &= -V'(u)\\ u& (-\infty) = u(+\infty) = 0 \end{aligned} \right. \] where \(u \in \mathbb{R}\), \(n \geq 2\) and \(V \in C^ 2(\mathbb{R}^ n \setminus \text{e}, \mathbb{R})\) is a potential with an absolute maximum at 0 and is such that \(V(x) \rightarrow -\infty\) as \(x \rightarrow \text{e}\), has at least \(n - 1\) geometrically distinct solutions under a rather complicated set of somewhat contrived conditions. The result is supposed to have application in finding homoclinic orbits for the potential in a Hamiltonian system.


58J32 Boundary value problems on manifolds
49Q20 Variational problems in a geometric measure-theoretic setting
70H05 Hamilton’s equations
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