American Institute of Mathematical Sciences

September  2016, 15(5): 1797-1807. doi: 10.3934/cpaa.2016015

A direct method of moving planes for fractional Laplacian equations in the unit ball

 1 Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

Received  October 2015 Revised  March 2016 Published  July 2016

In this paper, we employ a direct method of moving planes for the fractional Laplacian equation in the unit ball. Instead of using the conventional extension method introduced by Caffarelli and Silvestre [6], Chen, Li and Li developed a direct method of moving planes for the fractional Laplacian [8]. Inspired by this new method, in this paper we deal with the semilinear pseudo -differential equation in the unit ball directly. We first review key ingredients needed in the method of moving planes in a bounded domain, such as the narrow region principle for the fractional Laplacian. Then, by using this new method, we obtain the radial symmetry and monotonicity of positive solutions for some interesting semi-linear equations.
Citation: Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015
References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2st edition, Cambridge Studies in Advanced Mathematics, 116. Cambridge: Cambridge University, 2009. doi: 10.1017/CBO9780511809781. [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge: Cambridge University Press, 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] C. Brändle, E. Colorado, A. De Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc Royal Soc Edinburgh, A143 (2013), 39-71. [5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv in 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 A. 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. X. Chen, C. C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint, arXiv:1411.1697. [9] W. X. Chen, C. 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. [10] W. X. Chen, C. C. Li and B. Ou, Qualitative properities of solutions for an integral equation, Disc Cont Dyn Sys, 12 (2005), 347-354. [11] W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, preprint, arXiv:1404.1640. [12] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Mathematics, 1871 (2006), 1-43. doi: 10.1007/11545989_1. [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys, 68 (1979), 209-243. [14] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, \textbfA (1981), 369-402. [15] C. C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM Journal on Mathematical Analysis, 40 (2008), 1049-1057. doi: 10.1137/080712301. [16] L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, Journal of Differential Equations, 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04.008. [17] E. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull Sci Math, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [18] V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of systems with long-range interaction, Comm Non1 Sci Numer Simul, 11 (2006), 885-898. doi: 10.1016/j.cnsns.2006.03.005.

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References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2st edition, Cambridge Studies in Advanced Mathematics, 116. Cambridge: Cambridge University, 2009. doi: 10.1017/CBO9780511809781. [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge: Cambridge University Press, 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] C. Brändle, E. Colorado, A. De Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc Royal Soc Edinburgh, A143 (2013), 39-71. [5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv in 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 A. 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. X. Chen, C. C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint, arXiv:1411.1697. [9] W. X. Chen, C. 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. [10] W. X. Chen, C. C. Li and B. Ou, Qualitative properities of solutions for an integral equation, Disc Cont Dyn Sys, 12 (2005), 347-354. [11] W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, preprint, arXiv:1404.1640. [12] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Mathematics, 1871 (2006), 1-43. doi: 10.1007/11545989_1. [13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys, 68 (1979), 209-243. [14] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, \textbfA (1981), 369-402. [15] C. C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM Journal on Mathematical Analysis, 40 (2008), 1049-1057. doi: 10.1137/080712301. [16] L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, Journal of Differential Equations, 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04.008. [17] E. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull Sci Math, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [18] V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of systems with long-range interaction, Comm Non1 Sci Numer Simul, 11 (2006), 885-898. doi: 10.1016/j.cnsns.2006.03.005.
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