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Periodic solutions of a class of singular Hamiltonian systems. (English) Zbl 0648.34048
The author investigates the existence of periodic solutions, with a prescribed period, of the system of N second order differential equations $d\sp 2u/dt\sp 2=-V\sp 1(t,u)$ where $V\sp 1(t,.)$ denotes the gradient of the function V(t,.) defined on ${\bbfR}\sp N-\{0\}$. V(t,x) is supposed to be T-periodic int. With suitable restriction on V, basically related to its singular properties at $x=0$ and its behaviour at $\vert x\vert \to \infty$, theorems are proved pertaining to the existence of at least one non-constant T-periodic $C\sp 2$ solution, as well as the existence of infinitely many non-constant T-periodic $C\sp 2$ solutions. The proofs are heavily based on functional analysis, in particular of the nature and behaviour of the critical points. The paper mostly deals with $N>2$ when the set of singularities of V is simple.
Reviewer: N.D.Sengupta

##### MSC:
 34C25 Periodic solutions of ODE 70H05 Hamilton’s equations 34A34 Nonlinear ODE and systems, general
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##### References:
 [1] Ambrosetti, A.; Rabinowitz, P. H.: Dual variational methods in critical point theory and applications. J. funct. Analysis 14, 349-381 (1973) · Zbl 0273.49063 [2] Benci, V.: A geometrical index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations. Communs pure appl. Math. 34, 393-432 (1981) · Zbl 0447.34040 [3] Capozzi A., Greco C. & Salvatore A., Lagrangian systems in the presence of singularities, Proc. Am. math. Soc. (to appear). · Zbl 0664.34054 [4] Gordon, W. B.: Conservative dynamical systems involving strong forces. Trans. am. Math. soc. 204, 113-135 (1975) · Zbl 0276.58005 [5] Greco C., Periodic solutions of some nonlinear ODE with singular nonlinear part, Boll. Un. mat. ital. B (to appear) · Zbl 0644.34034 [6] Palais, S. R.: Lusternik-schnirelman theory on Banach manifolds. Topology 5, 115-132 (1966) · Zbl 0143.35203 [7] Rabinowitz, P. H.: Variational methods for nonlinear eigenvalue problems. Eigenvalue of nonlinear problems, 141-195 (1974) · Zbl 0278.35040 [8] Schwartz, J. T.: Nonlinear functional analysis. (1969) · Zbl 0203.14501