| Contents |
| We consider the planar N-centre problem with $\alpha$-homogeneous potential, $\alpha\in[1,2)$ and, for any $N\geq 3$ we prove the existence of infinitely many topologically distinct periodic solutions without collisions. Topologically distinct means that the solutions are not homotopy equivalent in the punctured plane. Indeed we provide sufficient ( and weak) conditions on the classes of the fundamental group of $ \mathbb R^{2}\setminus\{ c_{1}, \dots, c_{N} \}$ which ensure the existence of non collision solutions. In particular the solutions are allowed to self intersect and to design complicated paths around the singularities. The proof is based on variational techniques and relies on the so-called obstacle problem. |
|