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

Juncheng Wei; Ke Wu

doi:10.3934/dcds.2022044

Article Contents

2022, Volume 42, Issue 8: 4031-4050. Doi: 10.3934/dcds.2022044

This issue Previous Article Unfolding globally resonant homoclinic tangencies Next Article A fixed point theorem for twist maps

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

Juncheng Wei^1, and
Ke Wu^2, ,

1.
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada
2.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

^*Corresponding author: Ke Wu
^*Corresponding author: Ke Wu

Received: August 31, 2021

Early access: April 2022

Published: August 2022

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

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Abstract

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*} $

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

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