Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality

TL;DR

This paper analyzes qualitative properties of positive solutions to a Hardy--Sobolev type integral system, establishing decay rates, integrability criteria, symmetry, and monotonicity, and connecting these results to poly-harmonic systems.

Contribution

It provides new criteria for decay rates and integrability of solutions, extending classical results on elliptic systems to integral and poly-harmonic contexts.

Findings

Solutions decay with either fast or slow rates depending on integrability.

A criterion distinguishes integrable solutions based on asymptotic behavior.

Results include symmetry, boundedness, and optimal integrability of solutions.

Abstract

This article carries out a qualitative analysis on a system of integral equations of the Hardy--Sobolev type. Namely, results concerning Liouville type properties and the fast and slow decay rates of positive solutions for the system are established. For a bounded and decaying positive solution, it is shown that it either decays with the slow rates or the fast rates depending on its integrability. Particularly, a criterion for distinguishing integrable solutions from other bounded and decaying solutions in terms of their asymptotic behavior is provided. Moreover, related results on the optimal integrability, boundedness, radial symmetry and monotonicity of positive integrable solutions are also established. As a result of the equivalence between the integral system and a system of poly-harmonic equations under appropriate conditions, the results translate over to the corresponding poly-harmonic system. Hence, several classical results on semilinear elliptic systems are recovered and further generalized.