American Institute of Mathematical Sciences

Advanced
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

Meixia Dou 1,

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.
Keywords: monotonicity., the narrow region principle, radial symmetry, method of moving planes, The fractional Laplacian.
Mathematics Subject Classification: Primary: 35J60; Secondary: 53C21, 58J0.
Citation: Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & 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.  Google Scholar

[2]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge: Cambridge University Press, 1996.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. X. Chen, C. C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems,, preprint, ().   Google Scholar

[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.  Google Scholar

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys, 68 (1979), 209-243.  Google Scholar

[14]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, Mathematical Analysis and Applications, \textbfA (1981), 369-402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge: Cambridge University Press, 1996.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. X. Chen, C. C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems,, preprint, ().   Google Scholar

[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.  Google Scholar

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys, 68 (1979), 209-243.  Google Scholar

[14]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, Mathematical Analysis and Applications, \textbfA (1981), 369-402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[2]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[3]

Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235
[4]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445
[5]

Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082
[6]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462
[7]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[8]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
[9]

Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
[10]

Ryan Hynd, Francis Seuffert. On the symmetry and monotonicity of Morrey extremals. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5285-5303. doi: 10.3934/cpaa.2020238
[11]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[12]

Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237
[13]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[14]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[15]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
[16]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[17]

Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[18]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925
[19]

Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065
[20]

Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences