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Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity
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 |
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). $
References:
[1] |
G. Alberti and G. Bellettini,
A nonlocal anisotropic model for phase transitions, Math. Ann., 310 (1998), 527-560.
doi: 10.1017/S0956792598003453. |
[2] |
S. Barb, Topics in Geometric Analysis with Applications to Partial Differential Equations, Ph.D thesis, University of Missouri–Columbia, 2009. |
[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.
doi: 10.1016/j.aim.2012.03.032. |
[4] |
K. Bogdan, T. 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. |
[5] |
L. Brasco, E. 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. |
[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. |
[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. |
[8] |
W. X. Chen, C. 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. |
[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. |
[10] |
W. X. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific, 2020. |
[11] |
W. X. Chen, Y. 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. |
[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. |
[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. |
[14] |
E. Di Nezza, G. 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. |
[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. |
[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. |
[17] |
A. Iannizzotto, S. Mosconi and M. Squassina,
Global Hölder regularity for the fractional $ p $-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.
doi: 10.4171/RMI/921. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
S. Barb, Topics in Geometric Analysis with Applications to Partial Differential Equations, Ph.D thesis, University of Missouri–Columbia, 2009. |
[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.
doi: 10.1016/j.aim.2012.03.032. |
[4] |
K. Bogdan, T. 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. |
[5] |
L. Brasco, E. 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. |
[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. |
[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. |
[8] |
W. X. Chen, C. 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. |
[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. |
[10] |
W. X. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific, 2020. |
[11] |
W. X. Chen, Y. 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. |
[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. |
[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. |
[14] |
E. Di Nezza, G. 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. |
[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. |
[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. |
[17] |
A. Iannizzotto, S. Mosconi and M. Squassina,
Global Hölder regularity for the fractional $ p $-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.
doi: 10.4171/RMI/921. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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