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 \({1\over 2} |\dot q|^2 + V(q) = H,\) \(|q(t)| \longrightarrow \infty \quad {as} \quad t\longrightarrow\pm\infty\) \(\lim \limits_{t\to\pm\infty} {q(t)\over |q(t)|} = \theta_\pm.\)
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Received October 1998
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Felmer, P., Tanaka, K. Hyperbolic-like solutions for singular Hamiltonian systems. NoDEA, Nonlinear differ. equ. appl. 7, 43–65 (2000). https://doi.org/10.1007/PL00001422
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DOI: https://doi.org/10.1007/PL00001422