# American Institute of Mathematical Sciences

February  2021, 41(2): 537-552. 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, 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.  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. 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.  Google Scholar [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.  Google Scholar [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.   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. 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.  Google Scholar [12] W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019. Google Scholar [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.  Google Scholar [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.  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. 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.  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

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##### 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. 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.  Google Scholar [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.  Google Scholar [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.   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. 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.  Google Scholar [12] W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co., 2019. Google Scholar [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.  Google Scholar [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.  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. 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.  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
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