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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 & Applied Analysis, 2019, 18 (3) : 1073-1089. doi: 10.3934/cpaa.2019052
##### References:
  D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar  J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. Google Scholar  K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80.  Google Scholar  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.  Google Scholar  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar  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.  Google Scholar  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.  Google Scholar  W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1. Google Scholar  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.  Google Scholar  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.  Google Scholar  W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.  Google Scholar  T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. Google Scholar  Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015), Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar

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##### References:
  D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar  J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. Google Scholar  K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80.  Google Scholar  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.  Google Scholar  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar  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.  Google Scholar  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.  Google Scholar  W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1. Google Scholar  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.  Google Scholar  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.  Google Scholar  W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.  doi: 10.1007/s00220-006-0054-9.  Google Scholar  T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. Google Scholar  Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015), Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar
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