This paper is devoted to the existence and non-existence of positive solutions to the following negative power nonlinear integral equation related to the sharp reversed Hardy–Littlewood–Sobolev inequality: $$ f^{q-1}(x)= \int _{\varOmega }\frac{K(x)f(y)K(y)}{ \vert x-y \vert ^{n-\alpha }}\,dy+ \lambda \int _{\varOmega }\frac{G(x)f(y)G(y)}{ \vert x-y \vert ^{n-\alpha -\beta }}\,dy, \quad f\geq 0, x\in \overline{ \varOmega }, $$ where $0< q<1$, $\alpha >n$, $0<\beta <\alpha -n$, $\lambda \in \mathbb{R}$, Ω is a smooth bounded domain, $K(x)$, $G(x)$ are positive continuous functions in Ω̅. For $K\equiv G\equiv 1$, the existence and non-existence of positive solutions to the equation have been studied by Dou–Guo–Zhu (2019). In this paper we consider the existence and non-existence of positive solutions to the above integral equation with the general weight functions $K(x)$, $G(x)$.

中文翻译：

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具有权重的负功率非线性积分方程正解的存在性

本文致力于以下与负逆Hardy–Littlewood–Sobolev不等式有关的负幂非线性积分方程的正解的存在和不存在：$$ f ^ {q-1}（x）= \ int _ {\ varOmega} \ frac {K（x）f（y）K（y）} {\ vert xy \ vert ^ {n- \ alpha}} \，dy + \ lambda \ int _ {\ varOmega} \ frac {G （x）f（y）G（y）} {\ vert xy \ vert ^ {n- \ alpha-\ beta}} \，dy，\ quad f \ geq 0，x \ in \ overline {\ varOmega}， $$其中$ 0 <q <1 $，$ \ alpha> n $，$ 0 <\ beta <\ alpha -n $，$ \ lambda \ in \ mathbb {R} $，Ω是一个光滑有界域$ K（ x）$，$ G（x）$是Ω̅中的正连续函数。对于$ K \ equiv G \ equiv 1 $，Dou–Guo–Zhu（2019）研究了方程正解的存在和不存在。