Acta Mathematica Scientia

, Volume 39, Issue 1, pp 284–296 | Cite as

A Necessary Condition for Certain Integral Equations with Negative Exponents

  • Jiankai Xu (许建开)Email author
  • Zhong Tan (谭忠)
  • Weiwei Wang (王伟伟)
  • Zepeng Xiong (熊泽鹏)


This paper is devoted to studying the existence of positive solutions for the following integral system \(\left\{ {\begin{array}{*{20}{c}} {u\left( x \right) = \int_{{\mathbb{R}^n}} {{{\left| {x - y} \right|}^\lambda }{v^{ - q}}\left( y \right)dy,} } \\ {v\left( x \right) = \int_{{\mathbb{R}^n}} {{{\left| {x - y} \right|}^\lambda }{u^{ - p}}\left( y \right)dy,} } \end{array}} \right.p,q > 0,\lambda \in \left( {0,\infty } \right),n \geqslant 1\). It is shown that if (u, v) is a pair of positive Lebesgue measurable solutions of this integral system, then \(\frac{1}{{p - 1}} + \frac{1}{{q - 1}} = \frac{\lambda }{n}\), which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.

Key words

integral equations Lane-Emden system conformal invariance positive solutions existence 

2010 MR Subject Classification

45G05 45M99 


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Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Jiankai Xu (许建开)
    • 1
    Email author
  • Zhong Tan (谭忠)
    • 2
  • Weiwei Wang (王伟伟)
    • 3
  • Zepeng Xiong (熊泽鹏)
    • 4
  1. 1.College of Sciences; College of Computer ScienceHunan Agriculture UniversityChangshaChina
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenChina
  3. 3.College of Mathematics and Computer ScienceFuzhou UniversityFuzhouChina
  4. 4.The First Middle School of LonghuiLonghuiChina

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