Blowup analysis for integral equations on bounded domains

Consider the integral equation \begin{equation*} f^{q-1}(x)=\int_\Omega\frac{f(y)}{|x-y|^{n-\alpha}}dy,\ \ f(x)>0,\quad x\in \overline \Omega, \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a smooth bounded domain. For $1<\alpha <n$, the existence of energy maximizing positive solution in subcritical case $2<q<\frac{2n}{n+\alpha }$, and nonexistence of energy maximizing positive solution in critical case $q=\frac{2n}{n+\alpha }$ are proved in \cite{DZ2017}. For $\alpha >n$, the existence of energy minimizing positive solution in subcritical case $0<q<\frac{2n}{n+\alpha }$, and nonexistence of energy minimizing positive solution in critical case $q=\frac{2n}{n+\alpha }$ are also proved in \cite{DGZ2017}. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $q\to (\frac{2n}{n+\alpha }{)}^{+}$ (in the case of $1<\alpha <n$), and the blowup behaviour of energy minimizing positive solution as $q\to (\frac{2n}{n+\alpha }{)}^{-}$ (in the case of $\alpha >n$) are analyzed. We see that for $1<\alpha <n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving subcritical Sobolev exponent. But for $\alpha >n$, different phenomena appears.