LAUSR

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 ) = ∫ Ω K ( x ) f ( y ) K ( y ) | x − y | n − α d y + λ ∫ Ω G ( x ) f ( y ) G ( y ) | x − y | n − α − β d y , f ≥ 0 , x ∈ Ω ‾ , $$ f^{q-1}(x)= \int _{\varOmega }\frac{K(x)f(y)K(y)}{ \vert x-y \vert ^{n-\alpha }}\,dy+ \lambda \int _{\varOmega }\frac{G(x)f(y)G(y)}{ \vert x-y \vert ^{n-\alpha -\beta }}\,dy, \quad f\geq 0, x\in \overline{ \varOmega }, $$ where 0 < q < 1 $0< q<1$ , α > n $\alpha >n$ , 0 < β < α − n $0<\beta <\alpha -n$ , λ ∈ R $\lambda \in \mathbb{R}$ , Ω is a smooth bounded domain, K ( x ) $K(x)$ , G ( x ) $G(x)$ are positive continuous functions in Ω̅ . For K ≡ G ≡ 1 $K\equiv G\equiv 1$ , 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 ) $K(x)$ , G ( x ) $G(x)$ .