Positive solutions to  involving Wolff potentials

Huan-Zhen Chen; Zhongxue Lü

doi:10.3934/cpaa.2014.13.773

Article Contents

2014, Volume 13, Issue 2: 773-788. Doi: 10.3934/cpaa.2014.13.773

This issue Previous Article Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations Next Article A strongly singular parabolic problem on an unbounded domain

Positive solutions to involving Wolff potentials

Huan-Zhen Chen^1, and
Zhongxue Lü^2,

1.
School of Mathematical Sciences, Shandong Normal University, Jinan 250014
2.
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116

Received: April 2013

Revised: July 2013

Published: March 2014

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we consider the weighted integral system involving Wolff potentials in $R^{n}$: \begin{eqnarray} u(x) = R_1(x)W_{\beta, \gamma}(\frac{u^pv^q(y)}{|y|^\sigma})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{v^pu^q(y)}{|y|^\sigma})(x). \end{eqnarray} where $0< R(x) \leq C$, $1 < \gamma \leq 2$, $0\leq \sigma < \beta \gamma$, $n-\beta\gamma > \sigma(\gamma-1)$, $\gamma^{*}-1=\frac{n\gamma}{n-\beta\gamma+\sigma}-1\geq 1$. Due to the weight $\frac{1}{|y|^\sigma}$, we need more complicated analytical techniques to handle the properties of the solutions. First, we use the method of regularity lifting to obtain the integrability for the solutions of this Wolff type integral equation. Next, we use the modifying and refining method of moving planes established by Chen and Li to prove the radial symmetry for the positive solutions of related integral equation. Based on these results, we obtain the decay rates of the solutions of (0.1) with $R_1(x)\equiv R_2(x)\equiv 1$ near infinity. We generalize the results in the related references.

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

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