Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system
Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system
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DOI：
10.3934/dcds.2016.36.3791
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发表时间：
2016-03
影响因子：
1.1
通讯作者：
Yingshu Lü;Zhongxue Lü
中科院分区：
文献类型：
--
作者：
Yingshu Lü;Zhongxue Lü
本文研究加权 Hardy-Littlewood-Sobolev 型积分系统 \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} 和分数阶偏微分方程组 \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{数组} （2） \right. \end{方程} 这里$x \in R^n \setminus \{0\}$。由于 $0 < p, q < \infty$，我们需要更复杂的分析技术来处理 $0< p <1$ 或 $0< q <1$ 的情况。我们首先建立积分系统(1)和分数阶偏微分系统(2)的等价性。对于积分系统(1)，我们证明可积解是局部有界的。此外，我们还利用Chen-Li-Ou引入的积分形式移动平面的方法证明了正局部有界解是对称的并且关于某轴递减。由此可见，等价性意味着偏微分方程组的正解，也具有相应的性质。本文将其他作者先前获得的结果扩展到一般情况。
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.