*(English)*Zbl 0879.34043

Nonautonomous Hamiltonian systems of the form

where $I$ is the standard symplectic matrix

$x=({x}_{1},\cdots ,{x}_{2N})\in {\mathbb{R}}^{2N}$ and the Hamiltonian $H$ is a $T$-periodic (in the second variable) function, i.e. $H(x,t+T)=H(x,t)$, for all $(x,t)$ is considered. Precisely, for any given integer $p\ge 1$ the existence of multiple periodic (subharmonical) solutions of Eq. (1) $x\left(0\right)=x\left(pT\right)$ is investigated.

It is known, that under rather general assumptions on $H$ for any integer $p\ge 1$ the subharmonic problem (1) was shown to have a distinct solution ${X}_{\left(p\right)}$ and if $H$ has subquadratic or superquadratic growth at the origin and at infinity then the minimal period of ${X}_{\left(p\right)}$ tends towards infinity as $p\to +\infty $.

Some new multiplicity results are obtained for $H$ convex with subquadratic growth at the origin and at infinity. Roughly speaking, it is shown that there exist many subharmonics with exact minimal period $pT$ provided ${S}_{p}$ (the smallest prime factor of $p)$ is sufficiently large. Using a dual variational method the author answers the question whether the number of solutions of (1) with minimal period $pT$ tends towards infinity as $p\to +\infty $.