Periodic solutions to a forced Kepler problem in the plane

Given a smooth function $U(t,x)$, $T$-periodic in the first variable and satisfying $U(t,x)=\mathcal{O}(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })$ for some $\alpha \in (0,2)$as$\vert x \vert \to \infty$,weprovethattheforcedKeplerproblem$$\ddot{x}=-{\displaystyle \frac{x}{\mathrm{\mid }x{\mathrm{\mid }}^3}}+{\mathrm{\nabla }}_{x}U(t,x),x\in {\mathbb{R}}^2,$$hasageneralized$T$-periodic solution, according to the definition given in the paper [Boscaggin, Ortega, Zhao, \emph{Periodic solutions and regularization of a Kepler problem with time-dependent perturbation}, Trans. Amer. Math. Soc, 2018]. The proof relies on variational arguments.