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February  2021, 20(2): 835-865. doi: 10.3934/cpaa.2020293

Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian

1. 

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

2. 

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

3. 

Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

* Corresponding author

Received  July 2020 Revised  October 2020 Published  December 2020

Fund Project: The first author was supported by CHINA SCHOLARSHIP COUNCIL

We prove a Hopf's lemma in the point-wise sense for fractional $ p $-Laplacian. The essential technique is to prove $ (-\Delta)^s_p u(x) $ is uniformly bounded in the unit ball $ B_1\subset\mathbb{R}^n $, where $ u(x) = (1-|x|^2)^s_{+} $. Also we study the global Hölder continuity of bounded positive solutions for $ (-\Delta)^s_p u(x) = f(x,u). $

Citation: Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293
References:
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G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions, Math. Ann., 310 (1998), 527-560.  doi: 10.1017/S0956792598003453.  Google Scholar

[2]

S. Barb, Topics in Geometric Analysis with Applications to Partial Differential Equations, Ph.D thesis, University of Missouri–Columbia, 2009.  Google Scholar

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C. BjorlandL. Caffarelli and A. Figalli, Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.  doi: 10.1016/j.aim.2012.03.032.  Google Scholar

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K. BogdanT. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.  Google Scholar

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L. BrascoE. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional $ p $-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.  Google Scholar

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

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W. X. Chen and C. M. 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

[8]

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

[9]

W. X. Chen, C. M. Li and S. J. Qi, A Hopf lemma and regularity for fractional $ p $-Laplacians, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 3235-3252. Google Scholar

[10]

W. X. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific, 2020.

doi:10.1142//10550 Google Scholar

[11]

W. X. ChenY. Li and R. B. 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

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Y. G. Chen and B. Y. Liu, Symmetry and non-existence of solutions for fractional $ p $-Laplacian systems, Nonlinear Anal., 183 (2019), 303-322.  doi: 10.1016/j.na.2019.02.023.  Google Scholar

[13]

L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional $ p $-Laplacian, J. Differ. Equ., 263 (2017), 765-778.  doi: 10.1016/j.jde.2017.02.051.  Google Scholar

[14]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, B. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

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R. K. Getoor, First passage times for symmetric stable processes in space, T. Am. Math. Soc., 101 (1961), 75-90.  doi: 10.2307/1993412.  Google Scholar

[16]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885. doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[17]

A. IannizzottoS. Mosconi and M. Squassina, Global Hölder regularity for the fractional $ p $-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.  Google Scholar

[18]

H. Ishii and G. Nakamura, A class of integral equations and approximation of $ p $-Laplace equations, Calc. Var. Partial Differ. Equ., 37 (2010), 485-522.  doi: 10.1007/s00526-009-0274-x.  Google Scholar

[19]

L. Y. Jin and Y. Li, A Hopf's lemma and the boundary regularity for the fractional $ p $-Laplacian, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 1477-1495. doi: 10.3934/dcds.2019063.  Google Scholar

[20]

Z. Z. Li, On Some Problems for Fractional $ p $-Laplacian Operator and Nonlinear Elliptic Systems, Ph.D thesis, University of Chinese Academy of Sciences, 2020. Google Scholar

[21]

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

[22]

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

[23]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[24]

Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Commun. Partial Differ. Equ., 34 (2008), 765-784.  doi: 10.1080/03605300902892402.  Google Scholar

[25]

F. del Teso, D. Gómez-Castro and J. L. Vázquez, Three representations of the fractional $p$-Laplacian: semigroup, extension and Balakrishnan formulas, arXiv: 2010.06933. Google Scholar

[26]

L. Wu and W. X. Chen, The sliding methods for the fractional $ p $-Laplacian, Adv. Math., 361 (2020), 106933. doi: 10.1016/j.aim.2019.106933.  Google Scholar

show all references

References:
[1]

G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions, Math. Ann., 310 (1998), 527-560.  doi: 10.1017/S0956792598003453.  Google Scholar

[2]

S. Barb, Topics in Geometric Analysis with Applications to Partial Differential Equations, Ph.D thesis, University of Missouri–Columbia, 2009.  Google Scholar

[3]

C. BjorlandL. Caffarelli and A. Figalli, Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.  doi: 10.1016/j.aim.2012.03.032.  Google Scholar

[4]

K. BogdanT. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.  Google Scholar

[5]

L. BrascoE. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional $ p $-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.  Google Scholar

[6]

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

[7]

W. X. Chen and C. M. 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

[8]

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

[9]

W. X. Chen, C. M. Li and S. J. Qi, A Hopf lemma and regularity for fractional $ p $-Laplacians, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 3235-3252. Google Scholar

[10]

W. X. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific, 2020.

doi:10.1142//10550 Google Scholar

[11]

W. X. ChenY. Li and R. B. 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

[12]

Y. G. Chen and B. Y. Liu, Symmetry and non-existence of solutions for fractional $ p $-Laplacian systems, Nonlinear Anal., 183 (2019), 303-322.  doi: 10.1016/j.na.2019.02.023.  Google Scholar

[13]

L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional $ p $-Laplacian, J. Differ. Equ., 263 (2017), 765-778.  doi: 10.1016/j.jde.2017.02.051.  Google Scholar

[14]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, B. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[15]

R. K. Getoor, First passage times for symmetric stable processes in space, T. Am. Math. Soc., 101 (1961), 75-90.  doi: 10.2307/1993412.  Google Scholar

[16]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885. doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[17]

A. IannizzottoS. Mosconi and M. Squassina, Global Hölder regularity for the fractional $ p $-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.  Google Scholar

[18]

H. Ishii and G. Nakamura, A class of integral equations and approximation of $ p $-Laplace equations, Calc. Var. Partial Differ. Equ., 37 (2010), 485-522.  doi: 10.1007/s00526-009-0274-x.  Google Scholar

[19]

L. Y. Jin and Y. Li, A Hopf's lemma and the boundary regularity for the fractional $ p $-Laplacian, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 1477-1495. doi: 10.3934/dcds.2019063.  Google Scholar

[20]

Z. Z. Li, On Some Problems for Fractional $ p $-Laplacian Operator and Nonlinear Elliptic Systems, Ph.D thesis, University of Chinese Academy of Sciences, 2020. Google Scholar

[21]

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

[22]

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

[23]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

[24]

Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Commun. Partial Differ. Equ., 34 (2008), 765-784.  doi: 10.1080/03605300902892402.  Google Scholar

[25]

F. del Teso, D. Gómez-Castro and J. L. Vázquez, Three representations of the fractional $p$-Laplacian: semigroup, extension and Balakrishnan formulas, arXiv: 2010.06933. Google Scholar

[26]

L. Wu and W. X. Chen, The sliding methods for the fractional $ p $-Laplacian, Adv. Math., 361 (2020), 106933. doi: 10.1016/j.aim.2019.106933.  Google Scholar

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