Scattering parabolic solutions for the spatial N-centre problem

For the $N$-centre problem in the three dimensional space, $$\ddot{x}=-\sum _{i=1}^N\frac{m_i(x-c_i)}{\mathrm{\mid }x-c_i{\mathrm{\mid }}^{\alpha +2}},x\in {\mathbb{R}}^3\setminus \{c_1,\dots ,c_N\},$$ where $N\ge 2$, $m_i>0$ and $\alpha \in [1,2)$, we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions.