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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.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations
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##### References:
 [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 [4] \scAmbrosetti A. & \scCoti Zelati V., Perturbation of Hamiltonian systems with Keplerian potentials, preprint. · Zbl 0653.34032 [5] \scBahri A. & \scRabinowitz P. H., A minimax method for a class of Hamiltonian system with singular potentials, preprint (to appear). [6] Benci, V.; Giannoni, F., Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials,, J. diff. eqns, 82, 60-70, (1989) · Zbl 0689.34034 [7] Capozzi, A.; Salvatore, A.; Singh, Periodic solutions of Hamiltonian systems: the case of the singular potential, Proc. NATO-ASI, 207-216, (1986) · Zbl 0601.58031 [8] Capozzi, A.; Greco, C.; Salvatore, A., Lagrangian systems in presence of singularities, Proc. am. math. soc., 102, 125-130, (1988) · Zbl 0664.34054 [9] Coti Zelati, V., Remarks on dynamical systems with weak forces, Manuscripta math., 57, 417-424, (1987) · Zbl 0606.58039 [10] Coti Zelati, V., Dynamical systems with effective-like potential, Nonlinear analysis, 12, 209-222, (1988) · Zbl 0648.34050 [11] Degiovanni, M.; Giannoni, F., Periodic solutions of dynamical systems with Newtonian type potentials, Ann. scu. norm. sup. Pisa, 15, 467-494, (1988) · Zbl 0692.34050 [12] Degiovanni, M.; Giannoni, F.; Marino, A., Periodic solutions of dynamical systems with Newtonian type potentials, Atti accad. naz. lincei rc., LXXXI, 271-278, (1987) · Zbl 0667.70010 [13] Degiovanni, M.; Giannoni, F., Nonautonomous perturbations of Newtonian potential, Proc. nonlinear analysis and variational problems, (1986), Isola d’Elba · Zbl 0682.34031 [14] Gordon, W., A minimizing property of Keplerian orbits, Am. J. math., 99, 961-971, (1975) · Zbl 0378.58006 [15] Gordon, W., Conservative dynamical systems involving strong forces, Trans. am. math. soc., 204, 113-135, (1975) · Zbl 0276.58005 [16] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlinear analysis, 12, 259-269, (1988) · Zbl 0648.34048 [17] Solimi, S., On forced dynamical systems with a singularity of repulsive type, 14, 489-500, (1990) · Zbl 0708.34041
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