Nonhomogeneous nonlinear integral equations on bounded domains

This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form \begin{document}$ \begin{equation} u^{p-1}(x) = \int_{\Omega} \frac{u(y)}{|x-y|^{n-\alpha}} d y+\int_{\Omega} \frac{f(y)}{|x-y|^{n-\alpha}} d y, \ x \in \bar{\Omega}\nonumber \end{equation} $\end{document} where $ \frac{2n}{n+\alpha}\leq p < 2, $ $ 1 < \alpha < n $, $ n > 2, $ $ \Omega $ is a bounded domain in $ \mathbb R^{n} $. We show that under suitable assumptions on $ f, $ the integral equation admits a positive solution in $ L^{\frac{2n}{n+\alpha}}\left(\Omega\right) $. Our method combines the Ekeland variational principle, a blow-up argument and a rescaling argument which allows us to overcome the difficulties arising from the lack of Brezis-Lieb lemma in $ L^{\frac{2n}{n+\alpha}}(\Omega) $.