Liouville theorem for an integral system on the upper half space

Jingbo Dou; Ye Li

doi:10.3934/dcds.2015.35.155

Article Contents

2015, Volume 35, Issue 1: 155-171. Doi: 10.3934/dcds.2015.35.155

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Liouville theorem for an integral system on the upper half space

Jingbo Dou^1, and
Ye Li^2,

1.
School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100
2.
Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859

Received: January 2014

Revised: June 2014

Published: January 2015

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Abstract

In this paper we establish a Liouville type theorem for an integral system on the upper half space $\mathbb{R}_+^{n}$ \begin{equation*} \begin{cases} u(y)=\int_{\mathbb{R}^{n}_+}\frac{f(v(x))}{|x-y|^{n-\alpha}}dx,&\quad y\in\partial\mathbb{R}^{n}_+,\\ v(x)=\int_{\partial\mathbb{R}^{n}_+}\frac{g(u(y))}{|x-y|^{n-\alpha}}dy,&\quad x\in\mathbb{R}_+^{n}. \end{cases} \end{equation*} This integral system arises from the Euler-Lagrange equation corresponding to Hardy-Littlewood-Sobolev inequality on the upper half space. Under natural structure conditions on $f$ and $g$, we classify positive solutions to the above system basing on the method of moving sphere in integral forms and the Hardy-Littlewood-Sobolev inequality on the upper half space.

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Mathematics Subject Classification: Primary: 35B53; Secondary: 35B65.

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