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September  2020, 19(9): 4387-4399. doi: 10.3934/cpaa.2020200

Maximum principles for a fully nonlinear nonlocal equation on unbounded domains

Xiaoming He 1,, , Xin Zhao 2, and  Wenming Zou 2,

1. 

School of Science, Minzu University of China, Beijing 100081, China
2. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

*The corresponding author

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: X. He was supported by NSFC Grants (11771468, 11971027, 11271386), and W. Zou by NSFC Grants (11771234, 11926323, 11371212)

In this paper, we study equations involving fully nonlinear nonlocal operators
$ \mathcal {F}_{\alpha}(u(x)) = C_{n, \alpha}P.V.\int_{ \mathbb R^n}\frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}}dz = f(u(x)), \; \; \; x\in \mathbb R^n. $
We shall establish a maximum principle for anti-symmetric functions on any half space, and obtain key ingredients for proving the symmetry and monotonicity for positive solutions to the fully nonlinear nonlocal equations. Especially, a Liouville theorem is derived, which will be useful in carrying out the method of moving planes on unbounded domains for a variety of problems with fully nonlinear nonlocal operators.
Keywords: Maximum principle, Liouville theorem, fully nonlinear equation.
Mathematics Subject Classification: Primary: 35J60; 35J20; Secondary: 35R11; 35S15.
Citation: Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200
References:
[1]

F. Andreu, J. Mazon, J. Rossi and J. Toledo, Nonlocal Diffusion Problems, Vol. 165, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/165.

[2]

D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. 

[3]

C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.

[4]

K. Bogdan, T. Kulczycki and A. Nowak, gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. 

[5]

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

[6]

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

[7]

W. Chen, L. D. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal. Theory Meth. Appl., 121 (2015), 370-381.  doi: 10.1016/j.na.2014.11.003.

[8]

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.

[9]

W. Chen, C. Li and G. Li, Maximum priciple for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 29 pp. doi: 10.1007/s00526-017-1110-3.

[10]

W. Chen, C. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[12]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[13]

W. Chen and L. Wu, A maximum principle on unbounded domains and a Liouville theorem for fractional $p$-harmonic functions, preprint, arXiv: 1905.09986.

[14]

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

[15]

T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.

[16]

T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), Art. 1750018. doi: 10.1142/S0219199717500183.

[17]

E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora, (1979), 131–188.

[18]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[19]

X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differ. Equ., 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[20]

C. Kenig and W. Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math., 106 (1984), 689-702.  doi: 10.2307/2374291.

[21]

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

[22]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.

[23]

P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718.  doi: 10.3934/cpaa.2017082.

[24]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

F. Andreu, J. Mazon, J. Rossi and J. Toledo, Nonlocal Diffusion Problems, Vol. 165, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/165.

[2]

D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. 

[3]

C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.

[4]

K. Bogdan, T. Kulczycki and A. Nowak, gradient estimates for harmonic and $q$-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. 

[5]

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

[6]

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

[7]

W. Chen, L. D. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal. Theory Meth. Appl., 121 (2015), 370-381.  doi: 10.1016/j.na.2014.11.003.

[8]

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.

[9]

W. Chen, C. Li and G. Li, Maximum priciple for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 29 pp. doi: 10.1007/s00526-017-1110-3.

[10]

W. Chen, C. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[12]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[13]

W. Chen and L. Wu, A maximum principle on unbounded domains and a Liouville theorem for fractional $p$-harmonic functions, preprint, arXiv: 1905.09986.

[14]

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

[15]

T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.

[16]

T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), Art. 1750018. doi: 10.1142/S0219199717500183.

[17]

E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora, (1979), 131–188.

[18]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[19]

X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differ. Equ., 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[20]

C. Kenig and W. Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math., 106 (1984), 689-702.  doi: 10.2307/2374291.

[21]

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

[22]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part Ⅰ: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.

[23]

P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal., 16 (2017), 1707-1718.  doi: 10.3934/cpaa.2017082.

[24]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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