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

##### 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.  J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80. doi: 10.4064/sm-123-1-43-80.  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.  L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  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.  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.  W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1.  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.  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.  W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. 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.  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.  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.  Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.  T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015),  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.  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.  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.  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.  Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154  Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631  Wen Feng, Milena Stanislavova, Atanas Stefanov. 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