Existence of positive solutions to negative power nonlinear integral equations with weights

Abstract

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 dy + λ Ω G(x)f (y)G(y) |x-y| n-α-β dy, f ≥ 0, x ∈ Ω, where 0 < q < 1, α > n, 0 < β < α-n, λ ∈ R, Ω is a smooth bounded domain, K(x), G(x) are positive continuous functions in Ω. For K ≡ G ≡ 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).