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On a class of dynamical systems with singular potential. (English) Zbl 0744.34039
The authors consider the nonlinear differential system (*) \(-\ddot x=\nabla V(x)\), where \(x(t)\in C^ 2(R,R^ n)\), \(V\in C^ 1(R/\{0\},R)\) and \(V\) behaves like \(-k/| x|^ \alpha \) near the origin.
Their aim is to search for \(T\)-periodic solutions (with \(T\) prescribed) of (*). In literature this problem has been studied mainly by applying variational methods and considering two different situations: (i) \(1\leq \alpha < 2\), (ii) \(\alpha \geq 2\). In the present paper the particular case when \(\alpha=1\) is examined (assuming moreover that \(n=2\)); this case, as far as the authors know, was considered in one paper only [A. Ambrosetti and V. Coti Zelati, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 287-295 (1988; Zbl 0667.58055)]. Two new theorems concerning the existence and the properties of periodic solutions of (*) are proved.

34C25 Periodic solutions to ordinary differential equations
Full Text: DOI
[1] Ambrosetti, A.; Coti Zelati, V., Periodic solutions of singular dynamical systems, (), 1-10, NATO ASI series · Zbl 0757.70007
[2] Ambrosetti, A.; Coti Zelati, V., Solutions periodiques sans collision pour une classe de potentiels de type Keplerian, C.r. acad. sci. Paris, 305, 813-815, (1987) · Zbl 0639.34038
[3] Ambrosetti, A.; Coti Zelati, V., Noncollision orbits for a class of Keplerian-like potentials, Ann. inst. H. Poincaré analyse non linéaire, 5, 287-295, (1988) · Zbl 0667.58055
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[5] \scBahri A. & \scRabinowitz P. H., A minimax method for a class of Hamiltonian system with singular potentials, preprint (to appear).
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