Asymptotic behavior of positive singular solutions to fractional Hardy-Henon equations

Hui Yang Tsinghua University Wenming Zou Tsinghua University

Analysis of PDEs mathscidoc:1901.03004

In this paper, we study the asymptotic behavior of positive solutions of fractional Hardy-Henon equation with an isolated singularity at the origin. We give a classification of isolated singularities of positive solutions near x = 0. Further, we prove the asymptotic behavior of positive singular solutions. These results parallel those known for the Laplacian counterpart proved by Caffarelli, Gidas and Spruck (Caffarelli, Gidas and Spruck in Comm Pure Appl Math, 1981, 1989), but the technique is very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with a blow up (down) argument, the Kelvin transformation and uniqueness of solutions of related degenerate equations on semi spherical surface. We also investigate isolated singularities located at infinity of fractional Hardy-Henon equation.
Isolated singularities; asymptotic behavior; positive singular solutions; fractional Hardy-Henon equations
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@inproceedings{huiasymptotic,
  title={Asymptotic behavior of positive singular solutions to fractional Hardy-Henon equations},
  author={Hui Yang, and Wenming Zou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190109085156752309187},
}
Hui Yang, and Wenming Zou. Asymptotic behavior of positive singular solutions to fractional Hardy-Henon equations. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190109085156752309187.
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