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

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.

19 pages, author's accepted version including corrections to a few typographical errors

References in corpus (2)

• Sharp criteria of Liouville type for some nonlinear systems
• A degree theory approach for the shooting method

Cited by in corpus (6)

• Existence and non-existence results for the higher order Hardy-Hénon equation revisited
• Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality
• Sharp existence criteria for positive solutions of Hardy-Sobolev type systems
• A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients
• Asymptotic and optimal Liouville properties for Wolff type integral systems
• Shooting Method with Sign-Changing Nonlinearity