## Subharmonic solutions for some second-order differential equations with singularities.(English)Zbl 0787.34035

A scalar equation of the form $$u''+g(u)=e(t)(\equiv e(t+T))$$ is considered, where $$g$$ is a continuous function defined on an open interval $$(A,B)\subset R'$$, with some singular behavior at the boundary. The authors study the following two main cases:
(a) The interval $$(A,B)$$ is unbounded and $$g$$ is sublinear at infinity. In this case, via critical point theory, they prove the existence of a sequence of subharmonics whose amplitudes and minimal periods tend to infinity;
(b) The interval $$(A,B)$$ is bounded and the periodic forcing term $$e(t)$$ has minimal period $$T>0$$. In this case, using the generalized Poincaré- Birkhoff fixed-point theorem, they prove that for any positive integer $$m$$, 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.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 70F99 Dynamics of a system of particles, including celestial mechanics 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: