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Non-collision orbits for a class of Keplerian-like potentials. (English) Zbl 0667.58055
The authors consider the existence problem of T-periodic solutions x(t), x(t)\(\neq 0\), \(\forall t\in [0,T]\) (such solutions are called non- collision solutions) of the differential equation -ẍ\(=V'(x)\), where \(V\in C^ 1({\mathbb{R}}^ n\setminus \{0\},{\mathbb{R}})\) and V(x)\(\to -\infty\) as \(| x| \to 0\). The main result is the following. If there is an open, bounded set \(\Omega \subset {\mathbb{R}}^ n\) with smooth boundary \(\Gamma\) such that: (i) \(0\in \Omega\) and is star-shaped with respect to 0; (ii) max\(\{\) V(x): \(x\in {\mathbb{R}}^ n\setminus \{0\}\}\equiv V(\xi)\), \(\forall \xi \in \Gamma\); (iii) lim sup V(x)\(<V(\Gamma)\), then for sufficiently large T there is at least one non-collision solution x such that \(\{\) x(t)\(\}\) \(\not\subset \Gamma\). For an announcement see [the authors’ paper in C. R. Acad. Sci., Paris, Sér. I 305, 813-815 (1987; Zbl 0639.34038)].

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
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References:
[1] Ambrosetfi, A.; Zelati, V. Coti, Solutions périodiques sans collision pour une classe de potentiels de type keplerien, C.R. Acad. Sci. Paris, 305, 813-815, (1987) · Zbl 0639.34038
[2] A. Ambrosetti and V. Con Zelati, Critical Points with Lack of Compactness and Singular Dynamical Systems, Annali di Matematica Pura ed Applicata (to appear). · Zbl 0642.58017
[3] Ambrosetti, A.; Zelati, V. Coti, Periodic solutions of singular dynamical systems, Proceed. NATO ARW “Periodic Solutions of Hamiltonian Systems and Related Topics”, (1986), Il Ciocco Italy, (to appear) · Zbl 0757.70007
[4] Capozzi, A.; Greco, C.; Salvatore, A., Lagrangian systems in presence of singularities, (1985), preprint Universitá di Bari · Zbl 0664.34054
[5] Zelati, C. Coti, Remarks on dynamical systems with weak-forces, Manuscripta Math., Vol. 57, 417-424, (1987) · Zbl 0606.58039
[6] Degiovanni, M.; Giannoni, F.; Marino, A., Dynamical systems with Newtonian potentials, Proceed. NATO ARW “Periodic Solutions of Hamiltonian Systems and Related Topics”, (1986), Il Ciocco Italy, (to appear)
[7] Gordon, W., Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., Vol. 204, 113-135, (1975) · Zbl 0276.58005
[8] Gordon, W.; Minimizing, A., Property of Keplerian orbits, Am. Journ. of Math., Vol. 99, 961-971, (1977) · Zbl 0378.58006
[9] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, (1986), Università di Bari, preprint
[10] Klingenberg, W., Lectures on closed geodesics, (1978), Springer-Verlag Berlin · Zbl 0397.58018
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