Asymptotic and optimal Liouville properties for Wolff type integral systems

TL;DR

This paper investigates the asymptotic behavior and Liouville properties of positive solutions to nonlinear integral systems involving Hardy and Wolff potentials, providing existence results, decay rate characterizations, and applications to quasilinear systems.

Contribution

It establishes optimal existence, Liouville theorems, and a complete decay rate characterization for solutions of Wolff type integral systems, extending to quasilinear equations.

Findings

Solutions vanish at infinity with two principal decay rates

Decay rates are distinguished by an integrability criterion

Results apply to certain quasilinear systems

Abstract

This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact, the two rates can be fully distinguished by an integrability criterion. As an application, the results are shown to carry over to certain systems of quasilinear equations.