Periodic solutions for a class of symmetric and subquadratic Hamiltonian systems (Q1921088)

The authors consider a Hamiltonian system \(\dot z= JH_z(t,z)\), where \(H\in C^1 (\mathbb{R} \times \mathbb{R}^{2N}, \mathbb{R})\) is \(T\)-periodic in \(t\), even in \(z\) and subquadratic at \(z=0\). It is shown that under some additional hypotheses this system has infinitely many \(T\)-periodic solutions \(z_m\) which tend to zero as \(m\to\infty\). If moreover \(H\) is of the form \(H(t, z) = B(t)W(z)\) and \(T\) is the minimal period of \(B\), then \(T\) is also the minimal period of \(z_m\) for each \(m\). The proof uses a minimax argument and Krasnosel'skij's genus together with a Galerkin-type procedure (which is needed because the functional associated with the problem is strongly indefinite).