Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system

Yingshu  Lü; Zhongxue Lü

doi:10.3934/dcds.2016.36.3791

Article Contents

2016, Volume 36, Issue 7: 3791-3810. Doi: 10.3934/dcds.2016.36.3791

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Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system

Yingshu Lü^1, and
Zhongxue Lü^2,

1.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
2.
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116

Received: April 30, 2015

Revised: October 31, 2015

Published: May 2017

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Abstract

This paper is concerned with the properties of solutions for the weighted Hardy-Littlewood-Sobolev type integral system \begin{equation} \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy,\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy \end{array} \right. （1） \end{equation} and the fractional order partial differential system \begin{equation} \label{PDE} \left\{\begin{array}{ll} (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x). \end{array} （2） \right. \end{equation} Here $x \in R^n \setminus \{0\}$. Due to $0 < p, q < \infty$, we need more complicated analytical techniques to handle the case $0< p <1$ or $0< q <1$. We first establish the equivalence of integral system (1) and fractional order partial differential system (2). For integral system (1), we prove that the integrable solutions are locally bounded. In addition, we also show that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. Thus, the equivalence implies the positive solutions of the PDE system, also have the corresponding properties. This paper extends previous results obtained by other authors to the general case.

Keywords:

Weighted Hardy-Littlewood-Sobolev type integral system,
fractional order partial differential system,
method of moving planes,
symmetry.

Mathematics Subject Classification: 31B30, 35J48, 45E10, 45G05.

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