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Given a smooth function $U(t,x)$, $T$-periodic in the first variable and satisfying $U(t,x) = \mathcal{O}(\vert x \vert^{\alpha})$ for some $\alpha \in (0,2)$ 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.

For the spatial generalized $N$-centre problem $$ \ddot{x} = -\sum_{i=1}^{N} \frac{m_i (x - c_i)}{\vert x - c_i \vert^{\alpha+2}},\qquad x \in \mathbb{R}^3 \setminus \{c_1,\dots,c_N \}, $$ where $m_i > 0$ and $\alpha \in [1,2)$, we prove the existence of positive energy entire solutions with prescribed scattering angle. The proof relies on variational arguments, within an approximation procedure via (free-time) boundary value problems. A self-contained appendix describing a general strategy to rule out the occurrence of collisions is also included.