On isolated singularities of fractional semi-linear elliptic equations
Hui Yang, Wenming Zou · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2020
In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations (−\mathrm{\Delta })^{\sigma }u = u^{p} with an isolated singularity, where \sigma \in (0,1) and \frac{n}{n−2\sigma } < p < \frac{n + 2\sigma }{n−2\sigma } . We first use the blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When \sigma = 1 , this classification result has been proved by Gidas and Spruck (1981) [23], Caffarelli et al. (1989) [7].