# American Institute of Mathematical Sciences

February  2016, 36(2): 1125-1141. doi: 10.3934/dcds.2016.36.1125

## Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

 1 Department of Mathematical Sciences, Yeshiva University, New York, NY, 10033, United States 2 Department of Mathematics, Yeshiva University, New York, NY 10033 3 Department of Applied Mathematics, Northwestern Polytechnical University, Xian 710072, Shanxi, China 4 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, China

Received  July 2014 Revised  February 2015 Published  August 2015

In this paper, we consider the following system of pseudo-differential nonlinear equations in $R^n$ $$\left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u_i (x)= f_i( u_1(x), \cdots u_m(x)), & i=1, \cdots, m, \\ u_i \geq 0 , & i=1, \cdots, m, （1） \end{array} \right. \label{b1}$$ where $\alpha$ is any real number between $0$ and $2$.
We obtain radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.
To this end, we first establish the equivalence between (1) and the corresponding integral system $$\left\{\begin{array}{ll} u_i(x) = \int_{R^n} \frac{c_n}{|x-y|^{n-\alpha}} f_i( u_1(y), \cdots, u_m(y)), & i=1, \cdots, m, \\ u_i(x) \geq 0, & i=1, \cdots, m. \end{array} \right.$$ A new idea is introduced in the proof, which may hopefully be applied to many other problems. Combining this equivalence with the existing results on the integral system, we obtained much more general results on the qualitative properties of the solutions for (1).
Citation: Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [2] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. [3] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 1996. [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] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N. [6] 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. [7] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyna. Sys., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [8] G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. [9] W. Chen and C. Li, Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [10] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff.Equa.Dyn.Sys., 2010. [11] 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. [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [13] P. Constantin, Euler equations, navier-stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, 1871 (2006), 1-43. doi: 10.1007/11545989_1. [14] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [15] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [16] I. Capuzzo-Dolcetta and A. Cutri, On the Liouville property for sub-Laplacians, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 239-256. [17] L. Dupaigne and Y. Sire, A Liouville theorem for nonlocal elliptic equations, Symmetry for elliptic PDEs Contemp. Math., 528 (2010), 105-114. doi: 10.1090/conm/528/10417. [18] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [19] Y. Fang and J. Zhang, Nonexistence of positive solution for an integral equation on a half-space $R^n_+$, Comm. Pure and Applied Analysis, 12 (2013), 663-678. doi: 10.3934/cpaa.2013.12.663. [20] T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. [21] G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036. [22] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [23] E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Equations, 9 (1996), 465-479. [24] 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. [25] E. M. Stein, Singular Integrals and Differentiablity Properties of Funciotns, Princeton University Press, Princeton, 1970. [26] P. Stinga and C. Zhang, Harnack's inequality for fractional nonlocal equations, Disc. Cont. Dyn. Sys., 33 (2013), 3153-3170. doi: 10.3934/dcds.2013.33.3153. [27] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889. doi: 10.1016/j.cnsns.2006.03.005. [28] M. Zhu, Liouville theorems on some indefinite equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 649-661. doi: 10.1017/S0308210500021569. [29] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, arXiv:1401.7402.

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##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [2] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. [3] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 1996. [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] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N. [6] 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. [7] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyna. Sys., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [8] G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3. [9] W. Chen and C. Li, Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. [10] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff.Equa.Dyn.Sys., 2010. [11] 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. [12] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [13] P. Constantin, Euler equations, navier-stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, 1871 (2006), 1-43. doi: 10.1007/11545989_1. [14] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [15] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. [16] I. Capuzzo-Dolcetta and A. Cutri, On the Liouville property for sub-Laplacians, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 239-256. [17] L. Dupaigne and Y. Sire, A Liouville theorem for nonlocal elliptic equations, Symmetry for elliptic PDEs Contemp. Math., 528 (2010), 105-114. doi: 10.1090/conm/528/10417. [18] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [19] Y. Fang and J. Zhang, Nonexistence of positive solution for an integral equation on a half-space $R^n_+$, Comm. Pure and Applied Analysis, 12 (2013), 663-678. doi: 10.3934/cpaa.2013.12.663. [20] T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. [21] G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036. [22] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [23] E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Equations, 9 (1996), 465-479. [24] 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. [25] E. M. Stein, Singular Integrals and Differentiablity Properties of Funciotns, Princeton University Press, Princeton, 1970. [26] P. Stinga and C. Zhang, Harnack's inequality for fractional nonlocal equations, Disc. Cont. Dyn. Sys., 33 (2013), 3153-3170. doi: 10.3934/dcds.2013.33.3153. [27] V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889. doi: 10.1016/j.cnsns.2006.03.005. [28] M. Zhu, Liouville theorems on some indefinite equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 649-661. doi: 10.1017/S0308210500021569. [29] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, arXiv:1401.7402.
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