JOURNAL OF DIFFERENTIAL EQUATIONS 82, 670 (1989) Periodic Solutions of Prescribed Energy for a Class of Hamiltonian Systems with Singular Potentials* VIERI BENCI Istituto di Matematiche Applicate “U. Dini,” Universitb di Piss, Via Bonanno 25/B, 561OOPisa, Italy AND FABIO GIANNONI Dipartimento di Matematica, Vniversitci di Roma Tor Vergata, Via 0. Raimondo, 00173 Roma, Italy Received June 10, 1988 We study the existence of periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials. We found periodic solutions lying between two prescribed spheres. 0 1989 Academic press, 1~. 1. INTRODUCTION We consider the following O.D.E. 4”(t) + V’(q(t)) = 0, qE WR ~N\{o}), (1.1) where q”(t) is the second derivative of q(z), v’(x) is the gradient of the function I/ at x. and -1 V(x) = yp + W(x), WE C’(W, R), c? > 0. (1.2) Many recent papers concern the study of periodic solutions of (1.1). In particular, under suitable assumptions on tx and W, the existence of periodic solutions of prescribed period has been proved in [l-5, 7-11, 13-151. * Work supported by M.P.I. (4&60%, 1987). 60 0022-0396189 JE3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any lorm reserved.