用度理论打靶：若干加权多调和系统的分析
Shooting with degree theory: Analysis of some weighted poly-harmonic systems
摘要：
本文建立了一类半线性多调和方程组正整体解的存在性，这类方程组包括加权Hardy-Littlewood-Sobolev型方程和方程组.该方法实现了经典的打靶法，并通过拓扑度理论对其进行了改进。该方法的关键步骤是首先构造一个针对打靶法的目标映射，非退化条件保证了该映射的连续性。利用目标映射的连续性，利用拓扑变元证明了目标映射的零点存在性。该映射零点的存在性以及相应的Navier边值问题的不存在性定理意味着这类多调和方程组正解的存在性.
英文摘要：
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.
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