Classification of positive solutions to a Lane-Emden type integral system with negative exponents

Jingbo Dou; Fangfang  Ren; John  Villavert

doi:10.3934/dcds.2016094

Article Contents

2016, Volume 36, Issue 12: 6767-6780. Doi: 10.3934/dcds.2016094

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Classification of positive solutions to a Lane-Emden type integral system with negative exponents

1.
School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100
2.
School of National Fiscal Development, Central University of Finance and Economics, Beijing 100081
3.
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539

Received: December 31, 2015

Revised: March 31, 2016

Published: May 2017

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Abstract

In this paper, we classify the positive solutions to the following Lane-Emden type integral system with negative exponents \begin{equation*} \begin{cases} u(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau} u^{-p}(y)v^{-q}(y) \, dy, ~x\in \mathbb{R}^{n}, \\ v(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau}u^{-r}(y)v^{-s}(y) \, dy,~ x\in \mathbb{R}^{n}, \end{cases} \end{equation*}where $n \geq 1$ is an integer and $ \tau, p,q,r,s>0.$ Particularly, using an integral form of the method of moving spheres, we classify the positive solutions to the integral system whenever $$p+q=r+s=1 + 2n/\tau.$$ We also establish the non-existence of positive solutions under the condition $$\max\{p+q,r+s\} \leq 1 + 2n/\tau \,\text{ and }\, p + q + r + s < 2(1 + 2n/\tau).$$

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Mathematics Subject Classification: Primary: 45G15; Secondary: 35B65, 35B53.

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