Subharmonic and multiple subharmonic solutions for second order differential systems. (Q1413558)

The authors seek for subharmonic solutions of a system of second-order differential equations in \(\mathbb R^N\) of the form \[ \ddot u(t)+G'(u(t))=f(t) \quad \text{for } t\in \mathbb R, \tag{1} \] where \(N\) is a natural number, \(f\in C(\mathbb R,\mathbb R^N)\) is \(T\)-periodic, \(G\in C^2(\mathbb R^N,\mathbb R)\) is nonconvex, \(\ddot u={d^2u\over dt^2}\), and \(G'=\text{grad} G\). Here, a subharmonic solution of (1) means a periodic solution of (1) which is \(kT\)-periodic but not \(T\)-periodic for some natural \(k\). The existence of subharmonic solutions is proved by means of a variational argument and a Morse index theorem for minimax-type critical points. The authors show that, if \(f\) is contained in some subset of the space of \(T\)-periodic, continuous functions from \(\mathbb R\) into \(\mathbb R^N\), then there exist at least two \(kT\)-periodic but not \(T\)-periodic solutions of (1) for every sufficiently large prime number \(k\).