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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Liouville theorem for an integral system on the upper half space

Pages: 155 - 171, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.155

 
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Jingbo Dou - School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100, China (email)
Ye Li - Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, United States (email)

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.

Keywords:  Hardy-Littlewood-Sobolev inequality, integral system, Liouville type theorem, method of moving sphere.
Mathematics Subject Classification:  Primary: 35B53; Secondary: 35B65.

Received: January 2014;      Revised: June 2014;      Available Online: August 2014.

 References