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May  2019, 18(3): 1073-1089. doi: 10.3934/cpaa.2019052

The properties of positive solutions to semilinear equations involving the fractional Laplacian

 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

* Corresponding author

Received  December 2017 Revised  April 2018 Published  November 2018

Fund Project: The second author is supported by NSFC(No.11271166), NSF of Jiangsu Province(No. BK2010172), sponsored by Qing Lan Project.

Let
 $Ω$
be either a unit ball or a half space. Consider the following Dirichlet problem involving the fractional Laplacian
 \left\{ \begin{array}{*{35}{l}} \begin{align} & {{(-\Delta )}^{\frac{\alpha }{2}}}u=f(u),\ \ \text{in}\ \ \Omega , \\ & u=0, ~~~~~~~~~~~~~~~~~~~~ \text{in}\ \ {{\Omega }^{c}},\ \\ \end{align} & \ & {} \\\end{array} \right.~~~~(1)
where
 $α$
is any real number between
 zhongwenzy$and $
. Under some conditions on
 $f$
, we study the equivalent integral equation
 \begin{align}u(x) \ = \ \int{{}}_{ Ω}G(x, y)f(u(y))dy, \end{align}~~~~(2)
here
 $G(x, y)$
is the Green's function associated with the fractional Laplacian in the domain
 $Ω$
. We apply the method of moving planes in integral forms to investigate the radial symmetry, monotonicity and regularity for positive solutions in the unit ball. Liouville type theorems-non-existence of positive solutions in the half space are also deduced.
Citation: Rongrong Yang, Zhongxue Lü. The properties of positive solutions to semilinear equations involving the fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1073-1089. doi: 10.3934/cpaa.2019052
References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. [3] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80. [4] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990). doi: 10.1016/0370-1573(90)90099-N. [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [6] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [7] 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. [8] W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1. [9] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [10] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.  doi: 10.3934/cpaa.2005.4.1. [11] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. [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] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1. [14] P. Felmer and Y. Wang, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Comm. Cont. Math., 16 (2014), 1350023. doi: 10.1142/S0219199713500235. [15] Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9. [16] T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. [17] Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015), [18] 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. [19] 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. [20] 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. [21] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Radial symmetry of positive solutions to equations 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] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. [2] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. [3] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80. [4] J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990). doi: 10.1016/0370-1573(90)90099-N. [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [6] L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903. [7] 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. [8] W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1. [9] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [10] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.  doi: 10.3934/cpaa.2005.4.1. [11] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. [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] P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1. [14] P. Felmer and Y. Wang, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Comm. Cont. Math., 16 (2014), 1350023. doi: 10.1142/S0219199713500235. [15] Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9. [16] T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. [17] Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015), [18] 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. [19] 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. [20] 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. [21] R. Zhuo, W. Chen, X. Cui and Z. Yuan, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Discrete Contin.Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.
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