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

Abstract

<p style='text-indent:20px;'>In this paper, we study the local behavior of positive singular solutions to the equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} (-\Delta)^{\sigma}u = u^{\frac{n}{n-2\sigma}}\quad \;{\rm{in }}\;B_{1}\backslash\{0\} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ (-\Delta)^{\sigma} $\end{document}</tex-math></inline-formula> is the fractional Laplacian operator, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;\sigma&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \frac{n}{n-2\sigma} $\end{document}</tex-math></inline-formula> is the critical Serrin exponent. We show that either <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> can be extended as a continuous function near the origin or there exist two positive constants <inline-formula><tex-math id="M5">\begin{document}$ c_{1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ c_{2} $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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} </tex-math></disp-formula></p>