A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients

TL;DR

This paper characterizes the decay behavior of positive solutions for nonlinear integral systems with Wolff potentials and Hardy weights, linking fast decay to integrability and extending results to quasilinear Lane-Emden systems.

Contribution

It provides a complete characterization of fast decaying solutions in terms of integrability and establishes new decay properties for related quasilinear systems.

Findings

Fast decaying solutions are equivalent to integrable solutions.

Boundedness and optimal integrability of solutions are proven.

Decay properties are extended to quasilinear Lane-Emden systems.

Abstract

This paper examines the decay properties of positive solutions for a family of fully nonlinear systems of integral equations containing Wolf potentials and Hardy weights. This class of systems includes examples which are closely related to the Euler-Lagrange equations for several classical inequalities such as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular, a complete characterization of the fast decaying ground states in terms of their integrability is provided in that bounded and fast decaying solutions are shown to be equivalent to the integrable solutions. In generating this characterization, additional properties for the integrable solutions, such as their boundedness and optimal integrability, are also established. Furthermore, analogous decay properties for systems of quasilinear equations of the weighted Lane-Emden type are also obtained.