New Periodic Solutions for Some Planar $N+3$-Body Problems with Newtonian Potentials

For some planar Newtonian $N+3$-body problems, we use variational minimization methods to prove the existence of new periodic solutions satisfying that $N$ bodies chase each other on a curve, and the other 3 bodies chase each other on another curve. From the definition of the group action in equations $(3.1)-(3.3)$, we can find that they are new solutions which are also different from all the examples of Ferrario and Terracini (2004)$[22]$.