Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

Previous Article
Local rigidity of certain solvable group actions on tori
DCDS Home
This Issue
Next Article
Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation

February 2021, 41(2): 537-552. doi: 10.3934/dcds.2020268

Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

Lingwei Ma ^1, and Zhenqiu Zhang ^2,,

1.	School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

^* Corresponding author: Zhenqiu Zhang

Received December 2019 Revised May 2020 Published February 2021 Early access July 2020

In this paper, we first establish a narrow region principle for systems involving the fractional Laplacian in unbounded domains, which plays an important role in carrying on the direct method of moving planes. Then combining this direct method with the sliding method, we derive the monotonicity of bounded positive solutions to the following fractional Laplacian systems in unbounded Lipschitz domains

$ \Omega $

$ \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-\Delta\right)^{s}u& = &f(u,v), & \mbox{in}\ \ \Omega\,, \\[0.05cm] \left(-\Delta\right)^{t}v& = &g(u,v),& \mbox{in}\ \ \Omega\,, \\[0.05cm] u,\,v&\equiv&0, & \mbox{on}\ \ \mathbb{R}^{n}\setminus\Omega\,, \end{array}\right. \end{equation*} $

without any decay assumptions on the solution pair

$ (u,\,v) $

at infinity.

Keywords: Fractional Laplacian systems, unbounded Lipschitz domains, narrow region principle in unbounded domains, direct method of moving planes, sliding method, monotonicity.

Mathematics Subject Classification: Primary:35R11;Secondary:35B50, 35B99.

Citation: Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

References:

[1]	G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.
[2]	D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[3]	H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris, 29 (1993), 27-42.
[4]	H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494. doi: 10.1215/S0012-7094-96-08117-X.
[5]	H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:113.0.CO;2-6.
[6]	H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.
[7]	J. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N.
[8]	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.
[9]	L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[10]	W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domain, J. Funct. Anal., (2019).
[11]	W. Chen, C. 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.
[12]	W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019.
[13]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[14]	Z. Chen, C. Lin and W. Zou, Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105. doi: 10.1016/j.jfa.2013.08.021.
[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]	E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254. doi: 10.1007/s00208-008-0226-3.
[17]	S. Dipierro, N. Soave and E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326. doi: 10.1007/s00208-016-1487-x.
[18]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.
[19]	C. Li and Z. Wu, Radial symmetry for systems of fractional Laplacian, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 1567-1582. doi: 10.1016/S0252-9602(18)30832-4.
[20]	L. Ma and Z. Zhang, Symmetry of positive solutions for Choquard equations with fractional $p$-Laplacian, Nonlinear Anal., 182 (2019), 248-262. doi: 10.1016/j.na.2018.12.015.
[21]	A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659. doi: 10.1007/s00526-014-0727-8.

show all references

References:

[1]	G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.
[2]	D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[3]	H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris, 29 (1993), 27-42.
[4]	H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494. doi: 10.1215/S0012-7094-96-08117-X.
[5]	H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:113.0.CO;2-6.
[6]	H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.
[7]	J. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N.
[8]	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.
[9]	L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[10]	W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domain, J. Funct. Anal., (2019).
[11]	W. Chen, C. 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.
[12]	W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019.
[13]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.
[14]	Z. Chen, C. Lin and W. Zou, Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105. doi: 10.1016/j.jfa.2013.08.021.
[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]	E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254. doi: 10.1007/s00208-008-0226-3.
[17]	S. Dipierro, N. Soave and E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326. doi: 10.1007/s00208-016-1487-x.
[18]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.
[19]	C. Li and Z. Wu, Radial symmetry for systems of fractional Laplacian, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 1567-1582. doi: 10.1016/S0252-9602(18)30832-4.
[20]	L. Ma and Z. Zhang, Symmetry of positive solutions for Choquard equations with fractional $p$-Laplacian, Nonlinear Anal., 182 (2019), 248-262. doi: 10.1016/j.na.2018.12.015.
[21]	A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differential Equations, 52 (2015), 641-659. doi: 10.1007/s00526-014-0727-8.

[1]	Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235
[2]	Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015
[3]	Zhong-Qing Wang, Ben-Yu Guo, Yan-Na Wu. Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1019-1038. doi: 10.3934/dcdsb.2009.11.1019
[4]	Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74
[5]	Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082
[6]	Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462
[7]	Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2993-3020. doi: 10.3934/dcds.2020394
[8]	Elvira Zappale. A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains. Evolution Equations and Control Theory, 2017, 6 (2) : 299-318. doi: 10.3934/eect.2017016
[9]	Jacson Simsen, José Valero. Global attractors for $p$-Laplacian differential inclusions in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3239-3267. doi: 10.3934/dcdsb.2016096
[10]	Yunyun Hu. Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3723-3734. doi: 10.3934/cpaa.2020164
[11]	Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044
[12]	Esa V. Vesalainen. Rellich type theorems for unbounded domains. Inverse Problems and Imaging, 2014, 8 (3) : 865-883. doi: 10.3934/ipi.2014.8.865
[13]	Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[14]	Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800
[15]	Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281
[16]	Dario Cordero-Erausquin, Alessio Figalli. Regularity of monotone transport maps between unbounded domains. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297
[17]	Jian-Wen Sun. Nonlocal dispersal equations in domains becoming unbounded. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022076
[18]	Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028
[19]	Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[20]	Gyungsoo Woo, Young-Sam Kwon. Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains. Communications on Pure and Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Lingwei Ma Zhenqiu Zhang
on Google Scholar
Lingwei Ma Zhenqiu Zhang

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Lingwei Ma Zhenqiu Zhang
on Google Scholar
Lingwei Ma Zhenqiu Zhang

[Back to Top]