March  2020, 19(3): 1337-1349. doi: 10.3934/cpaa.2020065

Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation

1. 

School of mathematics and statistics, Anhui normal university, Wuhu, 241002, China

2. 

School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

* Corresponding author

Received  March 2019 Revised  July 2019 Published  November 2019

Fund Project: The work was supported by National Natural Science Foundation of China (11871096 and 11671308).

In this paper, we consider equations involving the fully nonlinear fractional order operator with homogeneous Dirichlet condition:
$ \begin{cases} F_\alpha(u)(x) = f(x,u,\nabla u) \ \mbox{in} \ \Omega,\\ u>0, \ \mbox{in}\ \Omega; \ u\equiv0, \ \mbox{in}\ \mathbb R^n\backslash\Omega, \end{cases} $
where
$ \Omega $
is a domain(bounded or unbounded) in
$ \mathbb R^n $
which is convex in
$ x_1- $
direction. By using some ideas of maximum principle, we prove that the solution is strictly increasing in
$ x_1- $
direction in the left half of
$ \Omega $
. Symmetry of solution is also proved. Meanwhile we obtain a Liouville type theorem on the half space
$ \mathbb R^n_+ $
.
Citation: Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065
References:
[1]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[2]

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.

[3]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.

[4]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[5]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18. doi: 10.1007/s00526-017-1110-3.

[6]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[7]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[8]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[9]

T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), 1750018, 12. doi: 10.1142/S0219199717500183.

[10]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.

[11]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[12]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[13]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.

[14]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.

[15]

Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.

[16]

C. LiZ. Wu and H. Xu, Maximum principles and bôcher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.

[17]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.

[18]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.

[19]

P. NiuL. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fractional Calculus and Applied Analysis, 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030.

[20]

Y. Wang and J. Wang, The method of moving planes for integral equation in an extremal case, J. Partial Differ. Equ., 29 (2016), 246-254.  doi: 10.4208/jpde.v29.n3.6.

show all references

References:
[1]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[2]

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.

[3]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.  doi: 10.1002/cpa.20274.

[4]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[5]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18. doi: 10.1007/s00526-017-1110-3.

[6]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[7]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[8]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[9]

T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), 1750018, 12. doi: 10.1142/S0219199717500183.

[10]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.

[11]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[12]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[13]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.

[14]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.

[15]

Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.

[16]

C. LiZ. Wu and H. Xu, Maximum principles and bôcher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.

[17]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.

[18]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.

[19]

P. NiuL. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fractional Calculus and Applied Analysis, 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030.

[20]

Y. Wang and J. Wang, The method of moving planes for integral equation in an extremal case, J. Partial Differ. Equ., 29 (2016), 246-254.  doi: 10.4208/jpde.v29.n3.6.

[1]

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

[2]

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

[3]

Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1

[4]

Weijun Zhou. A globally convergent BFGS method for symmetric nonlinear equations. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1295-1303. doi: 10.3934/jimo.2021020

[5]

Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021

[6]

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

[7]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

[8]

Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200

[9]

Xiaohui Yu. Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros. Communications on Pure and Applied Analysis, 2013, 12 (1) : 451-459. doi: 10.3934/cpaa.2013.12.451

[10]

Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

[12]

Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019

[13]

Dong-Hui Li, Xiao-Lin Wang. A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 71-82. doi: 10.3934/naco.2011.1.71

[14]

Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

[15]

Giovany Figueiredo, Marcelo Montenegro, Matheus F. Stapenhorst. A log–exp elliptic equation in the plane. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 481-504. doi: 10.3934/dcds.2021125

[16]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263

[17]

Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109

[18]

Qianzhong Ou. Nonexistence results for a fully nonlinear evolution inequality. Electronic Research Announcements, 2016, 23: 19-24. doi: 10.3934/era.2016.23.003

[19]

Luis Caffarelli, Luis Duque, Hernán Vivas. The two membranes problem for fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6015-6027. doi: 10.3934/dcds.2018152

[20]

Isabeau Birindelli, Stefania Patrizi. A Neumann eigenvalue problem for fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (249)
  • HTML views (78)
  • Cited by (1)

Other articles
by authors

[Back to Top]