The minimal period solution of Hamiltonian systems with even potentials

We consider the autonomous system $\ddot{q}+\mathrm{\nabla }V(q)=0$ in an arbitrary finite-dimensional Euclidean space. The potential is assumed to be even, nonnegative and of class $C^2$, to satisfy $V(x)=o(\mathrm{\mid }x{\mathrm{\mid }}^2)$ at the origin, and the Ambrosetti–Rabinowitz superquadratic inequality outside a ball. We prove that, for every prescribed $T>0$, the system has a nonconstant solution with $T$ being its minimal positive period.