Periodic solutions of Hamiltonian systems with anisotropic growth (Q1956571)

The authors consider the existence of \(T\)-periodic solutions to the Hamiltonian system \(\dot z = J\nabla H(z,t)\), \(z=(p_1,\dots,p_N,q_1,\dots,q_N)\in\mathbb R^{2N}\), where \(J\) is the standard symplectic matrix and \(H\) is \(T\)-periodic in \(t\). The main feature here is that \(H\), instead of the classical Ambrosetti-Rabinowitz superquadraticity condition, satisfies a condition which in particular allows superquadraticity in some variables, quadraticity in some other and subquadraticity in the remaining ones -- in a sense, (sub)quadraticity in \(p_j\) or \(q_j\) can be compensated by ``more superquadraticity'' in respectively \(q_j\) or \(p_j\). It is shown that under certain additional conditions there exists a \(T\)-periodic solution \(z\neq 0\) and, if \(H\) is even in \(z\), there are infinitely many such solutions. This generalizes some results in [\textit{P. L. Felmer}, J. Differ. Equations 102, No. 1, 188--207 (1993; Zbl 0781.34034)] and [\textit{P. Felmer} and \textit{Z.-Q. Wang}, Topol. Methods Nonlinear Anal. 12, No. 2, 207--226 (1998; Zbl 0931.35044)]. The arguments use abstract critical point theorems which may be found in these papers.