# American Institute of Mathematical Sciences

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 & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065
##### References:

show all references

##### References:
 [1] 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 [2] 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 [3] Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure & 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 & Management Optimization, 2021  doi: 10.3934/jimo.2021020 [5] Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2021  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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2010, 28 (2) : 845-863. doi: 10.3934/dcds.2010.28.845

2020 Impact Factor: 1.916