Periodic solutions of Lagrangian systems under small perturbations

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form d/dt(del Phi(u(t))) + V-u(t,u(t)) + lambda G(u)(t,u(t)) = 0, where Phi: R-n -> [0,infinity) is a G-function in the sense of Trudinger, V, G: R x R-n -> R are C-1-smooth, T-periodic in the time variable t and lambda is a real parameter. We prove the existence of a T-periodic solution u(lambda): R -> R-n for any sufficiently small |lambda|, and show that the found solutions converge to a T-periodic solution of the unperturbed system if lambda tends to 0. Let us stress that our theorem only makes a natural assumption on the potential V and no additional assumption on G except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz-Sobolev space setting, which bears several challenges.