# American Institute of Mathematical Sciences

March  2019, 39(3): 1595-1611. doi: 10.3934/dcds.2019071

## Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian

 1 School of mathematics and statistics, Huanghuai University, Zhumadian, Henan 463000, China 2 Department of Mathematics, Baylor University, Waco, TX 76798, USA

* Corresponding author: Ran Zhuo

Received  January 2018 Revised  April 2018 Published  December 2018

Fund Project: The first author is supported by NSFC grant 11701207 and Education Department of Henan Province grant 18B110011

In this paper, we consider the following Schrödinger systems involving pseudo-differential operator in
 $R^n$
 $\left\{ {\begin{array}{*{20}{l}}{{{( - \Delta )}^{\frac{\alpha }{2}}}u(x) = {u^{{\beta _1}}}(x){v^{{\tau _1}}}(x),}&{{\rm{in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\\{{{( - \Delta )}^{\frac{\gamma }{2}}}v(x) = {u^{{\beta _2}}}(x){v^{{\tau _2}}}(x),}& \ \ \ {{\rm{ in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\end{array}} \right.\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$
where
 $α$
and
 $γ$
are any number between 0 and 2,
 $α$
does not identically equal to
 $γ$
.
We employ a direct method of moving planes to partial differential equations (PDEs) (1). Instead of using the Caffarelli-Silvestre's extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-differential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.
In the proof, combining a new approach and the integral definition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at infinity. The new idea may hopefully be applied to many other problems.
Citation: Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
##### References:
 [1] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. [3] 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. [4] 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. [5] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [7] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274. [8] M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys., 38 (2018), 4603-4615. doi: 10.3934/dcds.2018201. [9] P. Constantin, Euler equations, navier-stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Math., 1871 (2006), 1-43. doi: 10.1007/11545989_1. [10] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Mathematical Journal, 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [11] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series on Diffreential Equatons & Dynamical Systerms, Volume 4, 2010. [12] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145 (1997), 547-564. doi: 10.2307/2951844. [13] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [14] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. [15] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. [16] 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. [17] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 1049-1057. doi: 10.1137/080712301. [18] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Linéaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [19] B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys., 38 (2018), 5339-5349. doi: 10.3934/dcds.2018235. [20] L. Ma and L. Zhao, Sharp thresholds of blow up and global existence for the coupled nonlinear Schrödinger system, J. Math. Phys., 49 (2008), 062103, 17 pp. doi: 10.1063/1.2939238. [21] 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. [22] D. Tang and Y. Fang, Method of sub-super solutions for fractional elliptic equations, Dis. Con. Dyn. Sys., 23 (2018), 3153-3165. doi: 10.3934/dcdsb.2017212. [23] 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. [24] R. Zhuo and F. Li, Liouville type theorems for Schrödinger systems in a half space, Science China Mathematics, 58 (2015), 179-196. doi: 10.1007/s11425-014-4925-9. [25] R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990. doi: 10.3934/cpaa.2014.13.977. [26] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Dis. Con. Dyn. Sys., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.

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##### References:
 [1] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. [3] 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. [4] 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. [5] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [7] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274. [8] M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys., 38 (2018), 4603-4615. doi: 10.3934/dcds.2018201. [9] P. Constantin, Euler equations, navier-stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Math., 1871 (2006), 1-43. doi: 10.1007/11545989_1. [10] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Mathematical Journal, 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [11] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series on Diffreential Equatons & Dynamical Systerms, Volume 4, 2010. [12] W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145 (1997), 547-564. doi: 10.2307/2951844. [13] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [14] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. [15] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. [16] 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. [17] C. Li and L. Ma, Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 1049-1057. doi: 10.1137/080712301. [18] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Linéaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [19] B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys., 38 (2018), 5339-5349. doi: 10.3934/dcds.2018235. [20] L. Ma and L. Zhao, Sharp thresholds of blow up and global existence for the coupled nonlinear Schrödinger system, J. Math. Phys., 49 (2008), 062103, 17 pp. doi: 10.1063/1.2939238. [21] 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. [22] D. Tang and Y. Fang, Method of sub-super solutions for fractional elliptic equations, Dis. Con. Dyn. Sys., 23 (2018), 3153-3165. doi: 10.3934/dcdsb.2017212. [23] 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. [24] R. Zhuo and F. Li, Liouville type theorems for Schrödinger systems in a half space, Science China Mathematics, 58 (2015), 179-196. doi: 10.1007/s11425-014-4925-9. [25] R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990. doi: 10.3934/cpaa.2014.13.977. [26] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Dis. Con. Dyn. Sys., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.
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