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In this paper, the author establishes the existence of positive entiresolutions to a general class of semilinear poly-harmonic systems, whichincludes equations and systems of the weighted Hardy--Littlewood--Sobolev type.The novel method used implements the classical shooting method enhanced bytopological degree theory. The key steps of the method are to first construct atarget map which aims the shooting method and the non-degeneracy conditionsguarantee the continuity of this map. With the continuity of the target map, atopological 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 thecorresponding Navier boundary value problem imply the existence of positivesolutions for the class of poly-harmonic systems.

Shooting with degree theory: Analysis of some weighted poly-harmonic systems