Blowup analysis for integral equations on bounded domains

TL;DR

This paper investigates the blowup behavior of solutions to certain integral equations on bounded domains as parameters approach critical values, revealing different phenomena depending on the relation between lpha and n.

Contribution

It provides a detailed analysis of the blowup behavior of energy maximizing and minimizing solutions near critical exponents for integral equations with different lpha regimes.

Findings

Blowup behavior similar to subcritical Sobolev exponent case for 1<lpha<n.

Distinct phenomena observed for lpha>n.

Results extend understanding of solution behavior near criticality.

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.