On the Hardy-Littlewood-Sobolev type systems

TL;DR

This paper surveys qualitative properties of Hardy-Littlewood-Sobolev type systems across critical, supercritical, and subcritical cases, introducing new insights, simplifications, and open problems for future research.

Contribution

It provides a comprehensive survey with new qualitative results, simplified methods, and extensions for HLS type systems in various regimes.

Findings

New qualitative properties of HLS systems

Simplified and extended methods for analysis

Open problems for future research

Abstract

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.