Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type

Xiaoqian Liu; Yutian Lei

doi:10.3934/dcds.2020018

Article Contents

2020, Volume 40, Issue 1: 467-489. Doi: 10.3934/dcds.2020018

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Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type

Xiaoqian Liu^1, and
Yutian Lei^2, ,

1.
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

^* Corresponding author: Yutian Lei
^* Corresponding author: Yutian Lei

Received: March 2019

Published: October 2019

This research is supported by the National Natural Science Foundation of China (11871278, 11671209)

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Abstract

This paper is concerned with the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system. Such a system is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality. The Serrin-type condition is critical for existence of positive solutions in $ L_{loc}^\infty(R^n \setminus \{0\}) $. When the Serrin-type condition does not hold, we prove the nonexistence by an iteration process. In addition, we find three pairs of radial solutions when the Serrin-type condition holds. One is singular, and the other two are integrable in $ R^n $ and decaying fast and slowly respectively.

Keywords:

Weighted Hardy-Littlewood-Sobolev inequality,
integral system,
existence of positive solution,
Serrin-type condition.

Mathematics Subject Classification: 45E10, 45G05, 45M20.

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