Nonhomogeneous nonlinear integral equations on bounded domains

This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form

$$u^{p-1}(x)={\int }_{\mathrm{\Omega }}\frac{u(y)}{|x-y{|}^{n-\alpha }}dy+{\int }_{\mathrm{\Omega }}\frac{f(y)}{|x-y{|}^{n-\alpha }}dy,x\in \overline{\mathrm{\Omega }}$$

where $\frac{2n}{n+\alpha }\le p<2,$ $1<\alpha <n$, $n>2,$ $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$. We show that under suitable assumptions on $f,$ the integral equation admits a positive solution in $L^{\frac{2n}{n+\alpha }}\left(\mathrm{\Omega }\right)$. 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 $L^{\frac{2n}{n+\alpha }}(\mathrm{\Omega })$.