The properties of positive solutions to an integral system involving Wolff potential

Huan Chen; Zhongxue Lü

doi:10.3934/dcds.2014.34.1879

Article Contents

2014, Volume 34, Issue 5: 1879-1904. Doi: 10.3934/dcds.2014.34.1879

This issue Previous Article A note on integrable mechanical systems on surfaces Next Article Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds

The properties of positive solutions to an integral system involving Wolff potential

Huan Chen^1, and
Zhongxue Lü^1,

1.
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116

Received: January 2013

Revised: June 2013

Published: May 2014

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: 45E10, 45G05, 45M05, 45M20.

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