Hyperbolic-like solutions for singular Hamiltonian systems

Abstract

We study the existence of unbounded solutions of singular Hamiltonian systems: \(\ddot q + \nabla V(q) = 0,\) where \(V(q) \sim -{1\over{|q|^\alpha}}\) is a potential with a singularity. For a class of singular potentials with a strong force \(\alpha>2\), we show the existence of at least one hyperbolic-like solutions. More precisely, for given \(H>0\) and \(\theta_+, \theta_-\in S^{N-1}\), we find a solution q(t) of (*) satisfying \({1\over 2} |\dot q|^2 + V(q) = H,\)\(|q(t)| \longrightarrow \infty \quad {as} \quad t\longrightarrow\pm\infty\)\(\lim \limits_{t\to\pm\infty} {q(t)\over |q(t)|} = \theta_\pm.\)