On a class of singular second-order Hamiltonian systems with infinitely many homoclinic solutions

TL;DR

This paper proves the existence of infinitely many homoclinic solutions at the origin for a class of singular second-order Hamiltonian systems using variational methods under the Strong-Force condition.

Contribution

It establishes the existence of infinitely many homoclinic orbits for a specific class of singular Hamiltonian systems, extending previous results.

Findings

Infinitely many homoclinic solutions at the origin.

Solutions are obtained under the Strong-Force condition.

Variational methods are effectively applied to singular systems.

Abstract

We show existence of infinitely many homoclinic orbits at the origin for a class of singular second-order Hamiltonian systems $$ \ddot{u} + V_u (t,u)=0\,,\quad -\infty < t < \infty\,. $$ We use variational methods under the assumption that\ $V(t,u)$\ satisfies the so-called "Strong-Force" condition.