March  2019, 39(3): 1269-1310. doi: 10.3934/dcds.2019055

Direct methods on fractional equations

1. 

Department of Mathematical Sciences, Yeshiva University, New York NY 10033, USA

2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, China

3. 

Department of Mathematical Sciences, Yeshiva University, USA

Received  January 2018 Revised  May 2018 Published  December 2018

Fund Project: The second author is partially supported Fundamental Research Funds for the Central Universities (lzujbky-2017-it53)

In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area.

Citation: Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055
References:
[1]

B. BarriosL. Del PezzoJ. Garcia-Melian and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, Revista Matematica Iberoamericana, 34 (2018), 195-220.  doi: 10.4171/RMI/983.  Google Scholar

[2]

K. BogdanT. Kulczycki and A. Nowak, Gradient estimates for harmonic and $ q $-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.   Google Scholar

[3]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[5]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure. Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.  Google Scholar

[6]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 4603-4615.  doi: 10.3934/dcds.2018201.  Google Scholar

[7]

W. ChenL. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Analysis, Theory, Methods & Appl, 121 (2015), 370-381.  doi: 10.1016/j.na.2014.11.003.  Google Scholar

[8]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.  Google Scholar

[9]

W. Chen and C. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667.  doi: 10.1002/cpa.3160480606.  Google Scholar

[10]

W. Chen and C. Li, A note on Kazdan-Warner conditions, J. Diff. Geom., 41 (1995), 259-268.  doi: 10.4310/jdg/1214456217.  Google Scholar

[11]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[12]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.  Google Scholar

[13]

W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

[14]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Cal. Var. & PDEs, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.  Google Scholar

[15]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.  Google Scholar

[16]

W. Chen, C. Li and Y. Li, A direct blow-up and rescaling argument on nonlocal elliptic equations, International J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.  Google Scholar

[17]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, a book to be published by World Scientific Publishing Co. 2017. Google Scholar

[18]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, CPAM, 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[19]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[20]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar

[21]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.  Google Scholar

[22]

W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, submitted to the same DCDS issue, 2017. Google Scholar

[23]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Equa., 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.  Google Scholar

[24]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Disc. Cont. Dyn. Sys. - Series A, 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.  Google Scholar

[25]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 3939-3953.   Google Scholar

[26]

M. Fall, Entire s-harmonic functions are affine, Proc. AMS, 144 (2016), 2587-2592.  doi: 10.1090/proc/13021.  Google Scholar

[27]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[28]

R. L. Frank and E. Lieb, Inversion positivityand the sharp Hardy-Littlewood-Sobolev inequality, Cal. Var. & PDEs, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.  Google Scholar

[29]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.  Google Scholar

[30]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.   Google Scholar

[31]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[32]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[33]

S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali Math. Pura et Appl., 195 (2016), 273-291.  doi: 10.1007/s10231-014-0462-y.  Google Scholar

[34]

T. JinY. 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.  Google Scholar

[35]

T. Jin and J. Xiong, A fractional Yemabe flow and some applications, J. reine angew. Math., 696 (2014), 187-223.  doi: 10.1515/crelle-2012-0110.  Google Scholar

[36]

J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Regional Conference Series in Mathematics, 57. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. doi: 10.1090/cbms/057.  Google Scholar

[37]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.  Google Scholar

[38]

Y. Lei, Asymptotic properties of positive solutions of the Hardy Sobolev type equations, J. Diff. Equa., 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.  Google Scholar

[39]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy Littlewood Sobolev system of integral equations, Cal. Var. & PDEs, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[40]

C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, June 20, 2018. Google Scholar

[41]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke. Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[42]

C. S. Lin, On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scula Norm. Sup. Pisa CI.Sci., 27 (1998), 107-130.   Google Scholar

[43]

B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 5339-5349.  doi: 10.3934/dcds.2018235.  Google Scholar

[44]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.  Google Scholar

[45]

G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.  Google Scholar

[46]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var. & PDEs, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.  Google Scholar

[47]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[48]

C. MaW. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math., 3 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[49]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[50]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. de Math. Pures et Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[51]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[52]

X. Xu and P. Yang, Remarks on prescribing Gaussian curvature, Trans. AMS, 336 (1993), 831-840.  doi: 10.1090/S0002-9947-1993-1087058-5.  Google Scholar

[53]

L. ZhangW. ChenC. Li and T. Cheng, A Liouville theorem for $ \alpha $-harmonic functions in $ \mathbb{R}^n_+ $, Disc. Cont. Dyn. Sys., 36 (2016), 1721-1736.  doi: 10.3934/dcds.2016.36.1721.  Google Scholar

[54]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.  Google Scholar

show all references

References:
[1]

B. BarriosL. Del PezzoJ. Garcia-Melian and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, Revista Matematica Iberoamericana, 34 (2018), 195-220.  doi: 10.4171/RMI/983.  Google Scholar

[2]

K. BogdanT. Kulczycki and A. Nowak, Gradient estimates for harmonic and $ q $-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.   Google Scholar

[3]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[5]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure. Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.  Google Scholar

[6]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 4603-4615.  doi: 10.3934/dcds.2018201.  Google Scholar

[7]

W. ChenL. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Analysis, Theory, Methods & Appl, 121 (2015), 370-381.  doi: 10.1016/j.na.2014.11.003.  Google Scholar

[8]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.  Google Scholar

[9]

W. Chen and C. Li, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math., 48 (1995), 657-667.  doi: 10.1002/cpa.3160480606.  Google Scholar

[10]

W. Chen and C. Li, A note on Kazdan-Warner conditions, J. Diff. Geom., 41 (1995), 259-268.  doi: 10.4310/jdg/1214456217.  Google Scholar

[11]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[12]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.  Google Scholar

[13]

W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

[14]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Cal. Var. & PDEs, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.  Google Scholar

[15]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.  Google Scholar

[16]

W. Chen, C. Li and Y. Li, A direct blow-up and rescaling argument on nonlocal elliptic equations, International J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.  Google Scholar

[17]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, a book to be published by World Scientific Publishing Co. 2017. Google Scholar

[18]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, CPAM, 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[19]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[20]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar

[21]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.  Google Scholar

[22]

W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, submitted to the same DCDS issue, 2017. Google Scholar

[23]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Equa., 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.  Google Scholar

[24]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Disc. Cont. Dyn. Sys. - Series A, 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.  Google Scholar

[25]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 3939-3953.   Google Scholar

[26]

M. Fall, Entire s-harmonic functions are affine, Proc. AMS, 144 (2016), 2587-2592.  doi: 10.1090/proc/13021.  Google Scholar

[27]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[28]

R. L. Frank and E. Lieb, Inversion positivityand the sharp Hardy-Littlewood-Sobolev inequality, Cal. Var. & PDEs, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.  Google Scholar

[29]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.  Google Scholar

[30]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.   Google Scholar

[31]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[32]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[33]

S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali Math. Pura et Appl., 195 (2016), 273-291.  doi: 10.1007/s10231-014-0462-y.  Google Scholar

[34]

T. JinY. 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.  Google Scholar

[35]

T. Jin and J. Xiong, A fractional Yemabe flow and some applications, J. reine angew. Math., 696 (2014), 187-223.  doi: 10.1515/crelle-2012-0110.  Google Scholar

[36]

J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Regional Conference Series in Mathematics, 57. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. doi: 10.1090/cbms/057.  Google Scholar

[37]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.  Google Scholar

[38]

Y. Lei, Asymptotic properties of positive solutions of the Hardy Sobolev type equations, J. Diff. Equa., 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.  Google Scholar

[39]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy Littlewood Sobolev system of integral equations, Cal. Var. & PDEs, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[40]

C. Li, Z. Wu and H. Xu, Maximum principles and Bocher type theorems, Proceedings of the National Academy of Sciences, June 20, 2018. Google Scholar

[41]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke. Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[42]

C. S. Lin, On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scula Norm. Sup. Pisa CI.Sci., 27 (1998), 107-130.   Google Scholar

[43]

B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 5339-5349.  doi: 10.3934/dcds.2018235.  Google Scholar

[44]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.  Google Scholar

[45]

G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.  Google Scholar

[46]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var. & PDEs, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.  Google Scholar

[47]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[48]

C. MaW. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math., 3 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[49]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[50]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. de Math. Pures et Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[51]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[52]

X. Xu and P. Yang, Remarks on prescribing Gaussian curvature, Trans. AMS, 336 (1993), 831-840.  doi: 10.1090/S0002-9947-1993-1087058-5.  Google Scholar

[53]

L. ZhangW. ChenC. Li and T. Cheng, A Liouville theorem for $ \alpha $-harmonic functions in $ \mathbb{R}^n_+ $, Disc. Cont. Dyn. Sys., 36 (2016), 1721-1736.  doi: 10.3934/dcds.2016.36.1721.  Google Scholar

[54]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.  Google Scholar

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