Existence of positive solutions to integral equations with weights

In this paper, we are concerned with the existence and non-existence of positive solutions to the following integral equation related to the sharp Hardy-Littlewood-Sobolev inequality f(q-1)(x) = integral(Omega) K(x)f(y)K(y)/vertical bar x - y vertical bar(n-alpha) dy + lambda integral(Omega) f(y)/vertical bar x - y vertical bar(n-alpha-beta) dy, f >= 0, x is an element of (Omega) over bar, where q > 1, 1 < alpha < n, 0 < beta < n - alpha, Omega is a smooth bounded domain, K(x) is positive continuous in (Omega) over bar. For K equivalent to const, beta = 1, the existence and non-existence of positive solutions to the equation has been studied by Dou and Zhu (2019). This paper is devoted to the case K(x) not equivalent to const. We want to point out that the weight function will bring difficulty when estimating the energy of the energy functional according to the integral equation. (C) 2019 Elsevier Ltd. All rights reserved.