Some results on the minimal period problem on nonconvex second order Hamiltonian systems (Q1284271)

For a differential system \(\ddot{x}+V'(x)=0, x\in {\mathbb R^N}\) with a \(C^2\)-smooth function, \(V(x)=h_0 x\cdot x +\widetilde V(x)\), \(\widetilde V\) is nonegative and of higher order at \(x=0\) with some restriction to the growth as \(x\to\infty\) the problem of existence and estimating from below of periodic solutions is studied. Previously \textit{P. H. Rabinowitz} proved [Commun. Pure Appl. Math. 31, 157-184 (1978; Zbl 0369.70017)] that if \(h_0 =0\) then the system possesses a nonconstant periodic solution with any period \(T>0\). This result is extended in the paper to the case when symmetric matrix \(h_0\) is positive semi-definite. The estimate is given in terms of indices of \(h_0\) which are defined in the paper. As an example, we present one of the assertions of the paper. Corollary 1.1. Suppose \(V\) satisfies the above-mentioned conditions. Then for every \(T\in (0, 2\pi/\sqrt{w_0})\) the system possesses a nonconstant \(T\)-periodic even solution with minimal period not less than \(T/(N+1)\), where \(w_0\) is the largest eigenvalue of \(h_0\).