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- W2930247670 abstract "In this paper, we study the asymptotic behavior of positive solutions of fractional Hardy-Henon equation $$ (-Delta)^sigma u = |x|^alpha u^p, ~~~~~~~ x in B_1 backslash {0} $$ with an isolated singularity at the origin, where $sigma in (0, 1)$ and the punctured unit ball $B_1 backslash {0} subset mathbb{R}^n$ with $n geq 2$. When $-2sigma < alpha < 2sigma$ and $frac{n+alpha}{n-2sigma} < p < frac{n+2sigma}{n-2sigma}$, we give a classification of isolated singularities of positive solutions near $x=0$. Further, we prove the asymptotic behavior of positive singular solutions as $xto 0$. 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 methods are 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 the uniqueness of solutions of related degenerate equations on $mathbb{S}^{n}_+$. We also investigate isolated singularities located at infinity of fractional Hardy-Henon equation." @default.
- W2930247670 created "2019-04-11" @default.
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- W2930247670 date "2019-03-31" @default.
- W2930247670 modified "2023-09-27" @default.
- W2930247670 title "Asymptotic behavior of positive singular solutions to fractional Hardy-Hénon equations" @default.
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