Periodic solutions of Lagrangian systems under small perturbations

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form [Formula: see text], where [Formula: see text] is a [Formula: see text]-function in the sense of Trudinger, [Formula: see text] are [Formula: see text]-smooth, [Formula: see text]-periodic in the time variable [Formula: see text] and [Formula: see text] is a real parameter. We prove the existence of a [Formula: see text]-periodic solution [Formula: see text] for any sufficiently small [Formula: see text], and show that the found solutions converge to a [Formula: see text]-periodic solution of the unperturbed system if [Formula: see text] tends to [Formula: see text]. Let us stress that our theorem only makes a natural assumption on the potential [Formula: see text] and no additional assumption on [Formula: see text] except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.