索波列夫空间
数学
纯数学
有界函数
数学分析
不平等
索波列夫不等式
类型(生物学)
作者
Yutian Lei,Zhongxue Lü
出处
期刊:Discrete and Continuous Dynamical Systems
[American Institute of Mathematical Sciences]
日期:2012-12-01
卷期号:33 (5): 1987-2005
被引量:15
标识
DOI:10.3934/dcds.2013.33.1987
摘要
This paper is concerned with the symmetry results for the
$2k$-order singular Lane-Emden type partial differential system
$$
\left\{\begin{array}{ll}
(-\Delta)^k(|x|^{\alpha}u(x))
=|x|^{-\beta} v^{q}(x),
\\
(-\Delta)^k(|x|^{\beta}v(x))
=|x|^{-\alpha} u^p(x),
\end{array}
\right.
$$
and the weighted Hardy-Littlewood-Sobolev type integral system
$$
\left \{
\begin{array}{l}
u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\
v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy.
\end{array}
\right.
$$
Here $x \in R^n \setminus \{0\}$. We first establish the
equivalence of this integral system and an fractional order
partial differential system, which includes the $2k$-order PDE
system above. For the integral system, we prove 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. In addition, we also show that the
integrable solutions are locally bounded. Thus, the equivalence
implies the positive solutions of the PDE system, particularly
including the higher integer-order PDE system, also have the
corresponding properties.
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