doi: 10.3934/dcds.2020268

Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains

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  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.
Citation: Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

H. BerestyckiL. 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.  Google Scholar

[5]

H. BerestyckiL. 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.  Google Scholar

[6]

H. BerestyckiL. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domain, J. Funct. Anal., (2019). Google Scholar

[11]

W. ChenC. 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

[12]

W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019. Google Scholar

[13]

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

[14]

Z. ChenC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. DipierroN. 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.  Google Scholar

[18]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

H. BerestyckiL. 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.  Google Scholar

[5]

H. BerestyckiL. 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.  Google Scholar

[6]

H. BerestyckiL. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domain, J. Funct. Anal., (2019). Google Scholar

[11]

W. ChenC. 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

[12]

W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019. Google Scholar

[13]

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

[14]

Z. ChenC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. DipierroN. 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.  Google Scholar

[18]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 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 & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015

[3]

Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082

[4]

Zhong-Qing Wang, Ben-Yu Guo, Yan-Na Wu. Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1019-1038. doi: 10.3934/dcdsb.2009.11.1019

[5]

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

[6]

Elvira Zappale. A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains. Evolution Equations & Control Theory, 2017, 6 (2) : 299-318. doi: 10.3934/eect.2017016

[7]

Jacson Simsen, José Valero. Global attractors for $p$-Laplacian differential inclusions in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3239-3267. doi: 10.3934/dcdsb.2016096

[8]

Yunyun Hu. Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3723-3734. doi: 10.3934/cpaa.2020164

[9]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020044

[10]

Esa V. Vesalainen. Rellich type theorems for unbounded domains. Inverse Problems & Imaging, 2014, 8 (3) : 865-883. doi: 10.3934/ipi.2014.8.865

[11]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201

[12]

Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051

[13]

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

[14]

Dario Cordero-Erausquin, Alessio Figalli. Regularity of monotone transport maps between unbounded domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297

[15]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[16]

Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028

[17]

Gyungsoo Woo, Young-Sam Kwon. Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains. Communications on Pure & Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135

[18]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23

[19]

Lijing Xi, Yuying Zhou, Yisheng Huang. A class of quasilinear elliptic hemivariational inequality problems on unbounded domains. Journal of Industrial & Management Optimization, 2014, 10 (3) : 827-837. doi: 10.3934/jimo.2014.10.827

[20]

Helmut Abels. Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 141-157. doi: 10.3934/dcdss.2010.3.141

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (59)
  • HTML views (179)
  • Cited by (0)

Other articles
by authors

[Back to Top]