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

  • *Corresponding author: Ke Wu

    *Corresponding author: Ke Wu

The research of J. Wei is partially supported by NSERC of Canada

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  • In this paper, we study the local behavior of positive singular solutions to the equation

    $ \begin{equation*} (-\Delta)^{\sigma}u = u^{\frac{n}{n-2\sigma}}\quad \;{\rm{in }}\;B_{1}\backslash\{0\} \end{equation*} $

    where $ (-\Delta)^{\sigma} $ is the fractional Laplacian operator, $ 0<\sigma<1 $ and $ \frac{n}{n-2\sigma} $ is the critical Serrin exponent. We show that either $ u $ can be extended as a continuous function near the origin or there exist two positive constants $ c_{1} $ and $ c_{2} $ such that

    $ \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*} $

    Mathematics Subject Classification: Primary: 35J60, 35J61; Secondary: 35A21.


    \begin{equation} \\ \end{equation}
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