Blowup analysis for integral equations on bounded domains

Consider the integral equation begin{equation*} f^{q-1}(x)=int_Omegafrac{f(y)}{|x-y|^{n-alpha}}dy, f(x)>0,quad xin overline Omega, end{equation*} where $Omegasubset mathbb{R}^n$ 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 $qto (frac{2n}{n+alpha})^+ $ (in the case of $1<alpha<n$), and the blowup behaviour of energy minimizing positive solution as $qto (frac{2n}{n+alpha})^-$ (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.