Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent

In this paper, we study the local behavior of positive singular solutions to the equation\begin{document}$ \begin{equation*} (-\Delta)^{\sigma}u = u^{\frac{n}{n-2\sigma}}\quad \;{\rm{in }}\;B_{1}\backslash\{0\} \end{equation*} $\end{document}where \begin{document}$ (-\Delta)^{\sigma} $\end{document} is the fractional Laplacian operator, \begin{document}$ 0<\sigma<1 $\end{document} and \begin{document}$ \frac{n}{n-2\sigma} $\end{document} is the critical Serrin exponent. We show that either \begin{document}$ u $\end{document} can be extended as a continuous function near the origin or there exist two positive constants \begin{document}$ c_{1} $\end{document} and \begin{document}$ c_{2} $\end{document} such that\begin{document}$ \begin{equation*} c_{1}|x|^{2\sigma-n}(-\ln{|x|})^{-\frac{n-2\sigma}{2\sigma}}\leq u(x)\leq c_{2}|x|^{2\sigma-n}(-\ln{|x|})^{-\frac{n-2\sigma}{2\sigma}}\quad\;{\rm{in }}\; B_{1}\backslash\{0\}. \end{equation*} $\end{document}