  In this paper, the author establishes the existence of positive entire
solutions to a general class of semilinear poly-harmonic systems, which
includes equations and systems of the weighted Hardy--Littlewood--Sobolev type.
The novel method used implements the classical shooting method enhanced by
topological degree theory. The key steps of the method are to first construct a
target map which aims the shooting method and the non-degeneracy conditions
guarantee the continuity of this map. With the continuity of the target map, a
topological argument is used to show the existence of zeros of the target map.
The existence of zeros of the map along with a non-existence theorem for the
corresponding Navier boundary value problem imply the existence of positive
solutions for the class of poly-harmonic systems.
