## Dynamical systems with effective-like potentials.(English)Zbl 0648.34050

In this paper it is studied the problem of existence of T-periodic solutions for the system of differential equations (V) $$-x''=\nabla_ xV(t,x)$$ where V(t,x) is assumed to be singular in the sense that $$V(t,x)\to +\infty$$ as $$| x| \to 0$$. Assuming: (i) V(t,x) is T- periodic in t; (ii) V(t,x), $$\nabla_ xV(t,x)\to 0$$ as $$| x| \to \infty$$; (iii) V(t,$$\lambda$$ x) is increasing in $$\lambda$$ for $$| \lambda |$$ small or large (uniformly in t) and (iv) V(t,x) satisfies a “strong force” condition (which concerns the behaviour of V(t,x) close to the singularity $$x=0)$$, it is proved the existence of at least one solution. In the case V does not depend on time, it is proved that it exists $$T_ 0>0$$ such that $$\forall T>T_ 0$$ there exists a T-periodic, non-trivial solution. Such results are proved applying Morse theory to the functional associated to (V), i.e. to the functional $$f(x)=\int^{T}_{0}| x'|^ 2dt-\int^{T}_{0}V(t,x)dt$$ and studying the topology of its level sets.
Reviewer: V.Coti Zelati

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37G99 Local and nonlocal bifurcation theory for dynamical systems

### Keywords:

singular potentials; Morse theory
Full Text:

### References:

 [1] {\scAmbrosetti}A. &{\scCoti Zelati}V., Solutions with minimal period for Hamiltonian systems in a potentials well, Annls. Inst. Henri Poincaré, Analyse non linéaire (to appear). [2] {\scAmbrosetti}A. &{\scCoti Zelati}V., Periodic solutions of dynamical systems with singular potential (in preparation). [3] Benci, V., Normal modes of a Lagrangian system in a potential well, Annls. inst. Henri Poincaré, analyse non linéaire, 1, 379-400, (1984) · Zbl 0561.58006 [4] Bott, R., Lectures on Morse theory, old and new, Bull. am. math. soc., 7, 331-358, (1982) · Zbl 0505.58001 [5] Capozzi, A.; Greco, G.; Salvatore, A., Lagrangian systems in presence of singularities, (1985), Bari University, preprint · Zbl 0664.34054 [6] 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 [7] Gordon, W., Conservative dynamical systems involving strong forces, Trans. am. math. soc., 204, 113-135, (1975) · Zbl 0276.58005 [8] Gordon, W., A minimizing property of Keplerian orbits, Am. J. math., 99, 961-971, (1975) · Zbl 0378.58006 [9] Greco, C., Periodic solutions of some ODE with singular nonlinear part, (1986), Bari University, preprint · Zbl 0644.34034 [10] {\scLazer}A. C. &{\scSolimini}S., On periodic solutions of nonlinear differential equations with singularity (to appear in Proc. A.M.S.).
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