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 $U(t,x) = \mathcal{O}(\vert x \vert^α)$ for some $α\in (0,2)$ as $\vert x \vert \to \infty$, we prove that the forced Kepler problem $$ \ddot x = - \dfrac{x}{|x|^3} + \nabla_x U(t,x),\qquad x\in {\mathbb{R}}^2, $$ has a generalized $T$-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.
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