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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:
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