Periodic solutions of a class of second order Hamiltonian systems. (English) Zbl 0847.34048

This paper considers the multiplicity result of \(T\)-periodic solutions for the differential equation \(\ddot x+ Ax+ \nabla F(x) =0\), where \(F\in C^1 (\mathbb{R}^n, \mathbb{R})\), \(A\neq 0\) is an \(n\times n\) symmetric real matrix which is not necessarily definite, and \(\nabla F\) denotes the gradient of \(F\). Roughly speaking, under the condition that \(F(x)\) is positively definite and \(F(sx)= s^\mu F(x)\) for a constant \(\mu>2\) and for all \(x\in \mathbb{R}^n\), the authors prove the existence of an infinity of nonconstant \(T\)-periodic solutions of the equation by applying the symmetric mountain pass theorem.


34C25 Periodic solutions to ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems