Blowup analysis for integral equations on bounded domains

Abstract

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 $\Omega\subset \mathbb{R}^n$ 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.