Periodic solutions for singular conservative systems (Q1336160)

The author looks for periodic solutions of the system \((*)\) \(\ddot q+ V' (q)= 0\), where \(q\in \mathbb{R}^N\) and \(V(x)\to -\infty\) as \(x\to 0\). This problem has been examined in the literature from two different points of view: a) given \(T>0\), one seeks for solutions of \((*)\) having \(T\) as a period; b) given \(h\in R\), one seeks for solutions of \((*)\) having \(h\) as energy. In this paper, on the contrary, periodic solutions having fixed mean potential energy are searched. By using some appropriate theorems, the author proves that there exist infinitely many such solutions of \((*)\).