Abstract
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.
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Supported by National Natural Science Foundation of China (11126148, 11501116, 11671086, 11871208), Natural Science Foundation of Hunan Province of China (2018JJ2159), the Project Supported by Scientific Research Fund of Hunan Provincial Education Department (16C0763) and the Education Department of Fujian Province (JA15063).
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Xu, J., Tan, Z., Wang, W. et al. A Necessary Condition for Certain Integral Equations with Negative Exponents. Acta Math Sci 39, 284–296 (2019). https://doi.org/10.1007/s10473-019-0121-x
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DOI: https://doi.org/10.1007/s10473-019-0121-x