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. AoH. ChanA. DelaTorreM. A. FontelosM. 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. CaffarelliB. 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. CaffarelliT. JinY. 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. DelaTorreM. del PinoM. 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. JinY. 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.

show all references

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. AoH. ChanA. DelaTorreM. A. FontelosM. 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. CaffarelliB. 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. CaffarelliT. JinY. 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. DelaTorreM. del PinoM. 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. JinY. 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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