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

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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

Rongrong Yang and Zhongxue Lü ^,

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.

Keywords: The fractional Laplacian, semi-linear equations, symmetry, monotonicity, regularity, Liouville theorem.

Mathematics Subject Classification: Primary: 35J60, 35B06, 35B09, 35B53, 45G10, 45M20.

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.

show all references

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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