Periodic solutions of Lagrangian systems under small perturbations

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form $\frac{d}{dt}(\nabla \mathrm{\Phi }(\dot{u}(t)))+V_u(t,u(t))+\lambda G_u(t,u(t))=0$, where $\mathrm{\Phi }:{ℝ}^n\to [0,\infty )$ is a ${𝒢}$-function in the sense of Trudinger, $V,G:ℝ\times {ℝ}^n\to ℝ$ 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 $u_{\lambda }:ℝ\to {ℝ}^n$ for any sufficiently small $\left|\lambda \right|$, 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.