# American Institute of Mathematical Sciences

March  2019, 39(3): 1545-1558. doi: 10.3934/dcds.2019067

## Liouville's theorem for a fractional elliptic system

 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China

* Corresponding author: Pengcheng Niu

Received  January 2018 Revised  May 2018 Published  December 2018

Fund Project: The authors are supported by the National Natural Science Foundation of China (No.11771354), China Postdoctoral Science Foundation (No.2017M613193)and Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University.

In this paper, we investigate the following fractional elliptic system
 $\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{\alpha /2}}u(x) = f(x){{v}^{q}}(x),&x\in {{R}^{n}}, \\ {{(-\Delta )}^{\beta /2}}v(x) = h(x){{u}^{p}}(x),&x\in {{R}^{n}}, \\\end{array} \right.$
where $1≤p, q < ∞$, $0 < α, β < 2$, $f(x)$ and $h(x)$ satisfy suitable conditions. Applying the method of moving planes, we prove monotonicity without any decay assumption at infinity. Furthermore, if $α = β$, a Liouville theorem is established.
Citation: Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
##### References:
 [1] W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601.  doi: 10.3934/dcds.2017242. [2] F. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2. [3] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, 121 Cambridge University Press, Cambridge, 1996. [4] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [6] 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. [7] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [8] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2011), 2497-2514.  doi: 10.3934/cpaa.2013.12.2497. [9] 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. [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, Y. Li and P. Ma, The fractional Laplacian, in press. [12] 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. [13] W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, accepted by Discrete Contin. Dyn. Syst. [14] T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154. [15] C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684. [16] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., 1871 (2004), 1-43.  doi: 10.1007/11545989_1. [17] Z. Dai, L. Cao and P. Wang, Liouville type theorems for the system of fractional nonlinear equations in $R^n_+$, J. Inequal. Appl., (2016), Paper No. 267, 17 pp. doi: 10.1186/s13660-016-1207-9. [18] J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Discrete Contin. Dyn. Syst., 38 (2018), 3939-3953.  doi: 10.3934/dcds.2018171. [19] D. Figueiredo, P. Lions and R. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. [20] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196. [21] H. Kaper and M. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17.  doi: 10.1007/978-1-4613-9608-6_1. [22] E. Leite ang M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, 2015, arXiv:1509.01267. [23] Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.  doi: 10.1007/s11425-016-0231-x. [24] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in ${R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874. [25] E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2011), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [26] P. Niu, L. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fract. Calc. Appl. Anal., 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030. [27] P. Pucci and V. Radulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-584. [28] 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. [29] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. [30] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 58 (2014), 133-154.  doi: 10.5565/PUBLMAT_58114_06. [31] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445. [32] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [33] V. Tarasov, G. Zaslavsky and M. George, Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-898.  doi: 10.1016/j.cnsns.2006.03.005. [34] 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. [35] L. Wu and P. Niu, Symmetry and Nonexistence of Positive Solutions to Fractional p-Laplacian Equations, to appeared in Discrete Contin. Dyn. Syst., 2018. [36] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence 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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##### References:
 [1] W. Ao, J. Wei and W. Yang, Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dyn. Syst., 37 (2017), 5561-5601.  doi: 10.3934/dcds.2017242. [2] F. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2. [3] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, 121 Cambridge University Press, Cambridge, 1996. [4] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [6] 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. [7] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [8] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure Appl. Anal., 12 (2011), 2497-2514.  doi: 10.3934/cpaa.2013.12.2497. [9] 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. [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, Y. Li and P. Ma, The fractional Laplacian, in press. [12] 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. [13] W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, accepted by Discrete Contin. Dyn. Syst. [14] T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Discrete Contin. Dyn. Syst., 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154. [15] C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684. [16] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., 1871 (2004), 1-43.  doi: 10.1007/11545989_1. [17] Z. Dai, L. Cao and P. Wang, Liouville type theorems for the system of fractional nonlinear equations in $R^n_+$, J. Inequal. Appl., (2016), Paper No. 267, 17 pp. doi: 10.1186/s13660-016-1207-9. [18] J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Discrete Contin. Dyn. Syst., 38 (2018), 3939-3953.  doi: 10.3934/dcds.2018171. [19] D. Figueiredo, P. Lions and R. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. [20] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196. [21] H. Kaper and M. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17.  doi: 10.1007/978-1-4613-9608-6_1. [22] E. Leite ang M. Montenegro, On positive viscosity solutions of fractional Lane-Emden systems, 2015, arXiv:1509.01267. [23] Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.  doi: 10.1007/s11425-016-0231-x. [24] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in ${R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874. [25] E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2011), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [26] P. Niu, L. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fract. Calc. Appl. Anal., 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030. [27] P. Pucci and V. Radulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., 3 (2010), 543-584. [28] 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. [29] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. [30] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 58 (2014), 133-154.  doi: 10.5565/PUBLMAT_58114_06. [31] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445. [32] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [33] V. Tarasov, G. Zaslavsky and M. George, Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-898.  doi: 10.1016/j.cnsns.2006.03.005. [34] 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. [35] L. Wu and P. Niu, Symmetry and Nonexistence of Positive Solutions to Fractional p-Laplacian Equations, to appeared in Discrete Contin. Dyn. Syst., 2018. [36] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence 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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