SUBHARMONIC SOLUTIONS FOR SOME 2ND-ORDER DIFFERENTIAL-EQUATIONS WITH SINGULARITIES

The existence of infinitely many subharmonic solutions is proved for the periodically forced nonlinear scalar equation u'' + g(u) = e(t), where g is a continuous function that is defined on a open proper interval (A, B) subset-of R. The nonlinear restoring field g is supposed to have some singular behaviour at the boundary of its domain. The following two main possibilities are analyzed: (a) The domain is unbounded and g is sublinear at infinity. In this case, via critical point theory, it is possible to prove the existence of a sequence of subharmonics whose amplitudes and minimal periods tend to infinity. (b) The domain is bounded and the periodic forcing term e(t) has minimal period T > 0. In this case, using the generalized Poincare-Birkhoff fixed point theorem, it is possible to show that for any m is-an-element-of N, there are infinitely many periodic solutions having mT as minimal period. Applications are given to the dynamics of a charged particle moving on a line over which one has placed some electric charges of the same sign.