On the Hardy-Littlewood-Sobolev type systems

Ze Cheng; Genggeng  Huang; Congming Li

doi:10.3934/cpaa.2016027

Article Contents

2016, Volume 15, Issue 6: 2059-2074. Doi: 10.3934/cpaa.2016027

This issue Previous Article Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data Next Article Robust control of a Cahn-Hilliard-Navier-Stokes model

On the Hardy-Littlewood-Sobolev type systems

1.
Department of Applied Mathematics, University of Colorado at Boulder, Colorado
2.
Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai
3.
Department of Applied Mathematics, University of Colorado at Boulder

Received: September 30, 2015

Revised: May 31, 2016

Published: October 2016

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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.

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Mathematics Subject Classification: Primary: 35J91.

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