Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space

Jingbo Dou; Ye Li

doi:10.3934/dcds.2018171

Article Contents

2018, Volume 38, Issue 8: 3939-3953. Doi: 10.3934/dcds.2018171

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Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space

Jingbo Dou^1, , and
Ye Li^2,

1.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China
2.
Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

* Corresponding author: Jingbo Dou
* Corresponding author: Jingbo Dou

Received: August 31, 2017

Revised: December 31, 2017

Published: May 2018

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Abstract

In this paper we mainly classify the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space $\mathbb{R}_+^{n}$, and also present some remarks on the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the whole space $\mathbb{R}^{n}$. Our main techniques are Kelvin transformation and the method of moving spheres in integral forms.

Keywords:

Logarithmic Hardy-Littlewood-Sobolev inequality,
extremal functions,
integral system,
Kelvin transformation,
method of moving spheres.

Mathematics Subject Classification: Primary: 45G15; Secondary: 31B10, 34A05.

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