Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation

Meng Qu; Ping Li; Liu Yang

doi:10.3934/cpaa.2020065

Article Contents

2020, Volume 19, Issue 3: 1337-1349. Doi: 10.3934/cpaa.2020065

This issue Previous Article Admissibility and polynomial dichotomies for evolution families Next Article A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities

Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation

Meng Qu^1,,
Ping Li^2, , and
Liu Yang^1,

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
^* Corresponding author

Received: February 28, 2019

Revised: June 30, 2019

Published: February 2020

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

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Abstract

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_+ $.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 35B33.

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