LAUSR

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