The properties of positive solutions to an integral system involving Wolff potential
The properties of positive solutions to an integral system involving Wolff potential
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DOI：
10.3934/dcds.2014.34.1879
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发表时间：
2013-10
影响因子：
1.1
通讯作者：
Huantong Chen;Zhongxue Lü
中科院分区：
文献类型：
--
作者：
Huantong Chen;Zhongxue Lü
In this paper, we consider the positive solutions of the following weighted integral system involving Wolff potential in $R^{n}$: $$ \left\{ \begin{array}{ll} u(x) = R_1(x)W_{\beta,\gamma}(\frac{v^q}{|y|^{\sigma}})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{u^p}{|y|^{\sigma}})(x). (0.1) \end{array} \right. $$ This system is helpful to understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\sigma=0$, it is difficult to handle the properties of the solutions since there is singularity at origin. First, we overcome this difficulty by modifying and refining the new method which was introduced to explore the integrability result establishes by Ma, Chen and Li, and obtain an optimal integrability. Second, we use the method of moving planes to prove the radial symmetry for the positive solutions of (0.1) when $R_{1}(x)\equiv R_{2}(x)\equiv 1$. Based on these results, by intricate analytical techniques, we obtain the estimate of the decay rates of those solutions near infinity.
In this paper, we consider the positive solutions of the following weighted integral system involving Wolff potential in $R^{n}$: $$ \left\{ \begin{array}{ll} u(x) = R_1(x)W_{\beta,\gamma}(\frac{v^q}{|y|^{\sigma}})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{u^p}{|y|^{\sigma}})(x). (0.1) \end{array} \right. $$ This system is helpful to understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\sigma=0$, it is difficult to handle the properties of the solutions since there is singularity at origin. First, we overcome this difficulty by modifying and refining the new method which was introduced to explore the integrability result establishes by Ma, Chen and Li, and obtain an optimal integrability. Second, we use the method of moving planes to prove the radial symmetry for the positive solutions of (0.1) when $R_{1}(x)\equiv R_{2}(x)\equiv 1$. Based on these results, by intricate analytical techniques, we obtain the estimate of the decay rates of those solutions near infinity.