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

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.

arXiv comment33 pages, final author version

References in corpus (5)

• Sharp criteria of Liouville type for some nonlinear systems
• Shooting with degree theory: Analysis of some weighted poly-harmonic systems
• A degree theory approach for the shooting method
• Sharp existence criteria for positive solutions of Hardy-Sobolev type systems
• Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type

Cited by in corpus (6)

• A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients
• Decay estimates and symmetry of finite energy solutions to elliptic systems in R^n
• Asymptotic and optimal Liouville properties for Wolff type integral systems
• A perturbative approach to non-degeneracy of the Lane-Emden system
• Minimal energy solutions to the fractional Lane-Emden system, I: Existence and singularity formation
• Nonlinear Inequalities with Double Riesz Potentials