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

TL;DR

This paper introduces a novel approach combining shooting method and topological degree theory to prove the existence of positive solutions for a broad class of weighted poly-harmonic systems, including Hardy-Littlewood-Sobolev type equations.

Contribution

It develops a new method that integrates topological degree theory with shooting techniques to establish existence results for complex poly-harmonic systems.

Findings

Proves existence of positive solutions for weighted poly-harmonic systems.

Introduces a target map with guaranteed continuity for the shooting method.

Establishes a non-existence theorem for certain boundary value problems.

Abstract

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.