For the $N$-centre problem in the three dimensional space, $$ \ddot x = -\sum_{ i=1}^{ N} \frac{ m_i \,(x-c_i)}{ \vert x - c_i \vert^{ \alpha+2}}, \qquad x \in \mathbb{ R}^3 \setminus \{ c_1,\ldots,c_N\}, $$ where $N \geq 2$, $m_i > 0$ 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.
Categories: math.CA math.AP
