The minimal periodic solutions for superquadratic autonomous Hamiltonian systems without the Palais-Smale condition (Q6635953)

"The authors prove the existence of periodic solutions with arbitrarily prescribed minimal period for even second order Hamiltonian \N\[\N\ddot{x}+V'(x)=0, \N\]\Nwith \(V\in C^2(\mathbb R^{2N},\mathbb R)\) satisfying conditions:\N\N\(\lim_{x\to 0}\frac{V(x)}{|x|^2}=0, \lim_{x\to \infty}\frac{V(x)}{|x|^2}=\infty, V(x)=V(-x),\)\N\N\( 0<V'(x)\cdot x\leq V""(x)x \cdot x\) for \(x\neq 0\).\N\NSimilarly, they prove the existence of periodic solutions of arbitrary minimal period for first order Hamiltonian systems with Hamiltonian only of \(C^1\)-class, under weak Nehari conditions instead of the Ambrosetti-Rabinowitz condition.\N\NIn order to do this, the authors extend the method of the Nehari manifold to the situation when the Nehari manifold is not a manifold."