A Liouville theorem for the subcritical Lane-Emden system

Ze Cheng; Genggeng Huang

doi:10.3934/dcds.2019058

Article Contents

2019, Volume 39, Issue 3: 1359-1377. Doi: 10.3934/dcds.2019058

This issue Previous Article Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians Next Article Non-existence of positive solutions for a higher order fractional equation

A Liouville theorem for the subcritical Lane-Emden system

Ze Cheng^1, and
Genggeng Huang^2, ,

1.
Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding author: Genggeng Huang
* Corresponding author: Genggeng Huang

Received: April 30, 2018

Published: December 2018

The first author is partially supported by NSF DMS-1405175. The second author is partially supported by NSFC-11401376 and the Scholarship of International Postdoctoral Exchange Fellowship Program

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Abstract

The Lane-Emden conjecture says that the subcritical Lane-Emden system admits no positive solution. In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy-type a priori estimate. The necessity of the condition we found can be easily checked. However, a major difficulty lies in the sufficiency. The proof is quite involving, but the benefit is that it reduces the longstanding problem to obtaining the a priori estimate of energy type.

Keywords:

Lane-Emden conjecture,
energy a priori estimates.

Mathematics Subject Classification: Primary: 35B09, 35B53, 35J05, 35J61.

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