The Rabinowitz minimal periodic solution conjecture

Abstract

In 1978, Rabinowitz proved the existence of a non-constant [Formula: see text]-periodic solution for nonlinear Hamiltonian systems on [Formula: see text] with Hamiltonian function being super-quadratic at the infinity and zero for any given [Formula: see text]. Since the minimal period of this solution may be [Formula: see text] for some positive integer [Formula: see text], he proposed the question whether there exists a solution with [Formula: see text] as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when [Formula: see text]. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it.