# American Institute of Mathematical Sciences

August  2022, 42(8): 4031-4050. doi: 10.3934/dcds.2022044

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

 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

Received  September 2021 Published  August 2022 Early access  April 2022

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

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
 $\begin{equation*} c_{1}|x|^{2\sigma-n}(-\ln{|x|})^{-\frac{n-2\sigma}{2\sigma}}\leq u(x)\leq c_{2}|x|^{2\sigma-n}(-\ln{|x|})^{-\frac{n-2\sigma}{2\sigma}}\quad\;{\rm{in }}\; B_{1}\backslash\{0\}. \end{equation*}$
Citation: Juncheng Wei, Ke Wu. Local behavior of solutions to a fractional equation with isolated singularity and critical Serrin exponent. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4031-4050. doi: 10.3934/dcds.2022044
##### References:
 [1] P. Aviles, On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J., 32 (1983), 773-791.  doi: 10.1512/iumj.1983.32.32051. [2] P. Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192.  doi: 10.1007/BF01210610. [3] W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. del Mar González and J. Wei, On higher dimensional singularities for the fractional Yamabe problem: A non-local Mazzeo-Pacard program, Duke Math. J., 168 (2019), 3297-3411.  doi: 10.1215/00127094-2019-0034. [4] W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. del Mar González and J. Wei, Existence of positive weak solutions for fractional Lane-Emden equations with prescribed singular sets, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 149, 25 pp. [5] W. Ao, H. Chan, M. del Mar González, A. Hyder and J. Wei, Removability of singularities and superharmonicity for some fractional Laplacian equations, preprint, 2020, arXiv: 2001.11683v2. [6] M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539.  doi: 10.1007/BF01243922. [7] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [8] L. Caffarelli, T. Jin, Y. Sire and J. Xiong, Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities, Arch. Ration. Mech. Anal., 213 (2014), 245-268.  doi: 10.1007/s00205-014-0722-4. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [10] H. Chan and A. DelaTorre, An analytic construction of singular solutions related to a critical Yamabe problem, Comm. Partial Differential Equations, 45 (2020), 1621-1646.  doi: 10.1080/03605302.2020.1784209. [11] H. Chan and A. DelaTorre, Singular solutions of a critical fractional Yamabe problem, Work in progress. [12] C.-C. Chen and C. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal., 9 (1999), 221-246.  doi: 10.1007/BF02921937. [13] H. Chen and A. Quaas, Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results, J. Lond. Math. Soc., 97 (2018), 196-221.  doi: 10.1112/jlms.12104. [14] A. DelaTorre, M. del Pino, M. González and J. Wei, Delaunay-type singular solutions for the fractional Yamabe problem, Math. Ann., 369 (2017), 597-626.  doi: 10.1007/s00208-016-1483-1. [15] M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918. [16] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406. [17] T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.  doi: 10.4171/JEMS/456. [18] Y. Li and J. Bao, Local behavior of solutions to fractional Hardy-H$\acute{e}$non equations with isolated singularity, Ann. Mat. Pura Appl., 198 (2019), 41-59.  doi: 10.1007/s10231-018-0761-9. [19] R. Mazzeo and F. Pacard, A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Differential Geom., 44 (1996), 331-370. [20] F. Pacard, Existence and convergence of positive weak solutions of $-\Delta u = u^{\frac{n}{n-2}}$ in bounded domains of $\mathbb{R}^{n}, n\geq3$., Calc. Var. Partial Differential Equations, 1 (1993), 243-265.  doi: 10.1007/BF01191296. [21] N. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, 1968. [22] H. Yang and W. Zou, Exact asymptotic behavior of singular positive solutions of fractional semi-linear elliptic equations, Proc. Amer. Math. Soc., 147 (2019), 2999-3009.  doi: 10.1090/proc/14448. [23] H. Yang and W. Zou, On isolated singularities of fractional semi-linear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 403-420.  doi: 10.1016/j.anihpc.2020.07.003.

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##### References:
 [1] P. Aviles, On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J., 32 (1983), 773-791.  doi: 10.1512/iumj.1983.32.32051. [2] P. Aviles, Local behavior of solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192.  doi: 10.1007/BF01210610. [3] W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. del Mar González and J. Wei, On higher dimensional singularities for the fractional Yamabe problem: A non-local Mazzeo-Pacard program, Duke Math. J., 168 (2019), 3297-3411.  doi: 10.1215/00127094-2019-0034. [4] W. Ao, H. Chan, A. DelaTorre, M. A. Fontelos, M. del Mar González and J. Wei, Existence of positive weak solutions for fractional Lane-Emden equations with prescribed singular sets, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 149, 25 pp. [5] W. Ao, H. Chan, M. del Mar González, A. Hyder and J. Wei, Removability of singularities and superharmonicity for some fractional Laplacian equations, preprint, 2020, arXiv: 2001.11683v2. [6] M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539.  doi: 10.1007/BF01243922. [7] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304. [8] L. Caffarelli, T. Jin, Y. Sire and J. Xiong, Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities, Arch. Ration. Mech. Anal., 213 (2014), 245-268.  doi: 10.1007/s00205-014-0722-4. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [10] H. Chan and A. DelaTorre, An analytic construction of singular solutions related to a critical Yamabe problem, Comm. Partial Differential Equations, 45 (2020), 1621-1646.  doi: 10.1080/03605302.2020.1784209. [11] H. Chan and A. DelaTorre, Singular solutions of a critical fractional Yamabe problem, Work in progress. [12] C.-C. Chen and C. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geom. Anal., 9 (1999), 221-246.  doi: 10.1007/BF02921937. [13] H. Chen and A. Quaas, Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results, J. Lond. Math. Soc., 97 (2018), 196-221.  doi: 10.1112/jlms.12104. [14] A. DelaTorre, M. del Pino, M. González and J. Wei, Delaunay-type singular solutions for the fractional Yamabe problem, Math. Ann., 369 (2017), 597-626.  doi: 10.1007/s00208-016-1483-1. [15] M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918. [16] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406. [17] T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.  doi: 10.4171/JEMS/456. [18] Y. Li and J. Bao, Local behavior of solutions to fractional Hardy-H$\acute{e}$non equations with isolated singularity, Ann. Mat. Pura Appl., 198 (2019), 41-59.  doi: 10.1007/s10231-018-0761-9. [19] R. Mazzeo and F. Pacard, A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Differential Geom., 44 (1996), 331-370. [20] F. Pacard, Existence and convergence of positive weak solutions of $-\Delta u = u^{\frac{n}{n-2}}$ in bounded domains of $\mathbb{R}^{n}, n\geq3$., Calc. Var. Partial Differential Equations, 1 (1993), 243-265.  doi: 10.1007/BF01191296. [21] N. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, 1968. [22] H. Yang and W. Zou, Exact asymptotic behavior of singular positive solutions of fractional semi-linear elliptic equations, Proc. Amer. Math. Soc., 147 (2019), 2999-3009.  doi: 10.1090/proc/14448. [23] H. Yang and W. Zou, On isolated singularities of fractional semi-linear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 403-420.  doi: 10.1016/j.anihpc.2020.07.003.
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