Maximum principles for a fully nonlinear nonlocal equation on unbounded domains

Xiaoming He; Xin Zhao; Wenming Zou

doi:10.3934/cpaa.2020200

Article Contents

2020, Volume 19, Issue 9: 4387-4399. Doi: 10.3934/cpaa.2020200

This issue Previous Article Large time behavior of ODE type solutions to parabolic

$ p $

-Laplacian type equations Next Article Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials

Maximum principles for a fully nonlinear nonlocal equation on unbounded domains

1.
School of Science, Minzu University of China, Beijing 100081, China
2.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

^*The corresponding author
^*The corresponding author

Received: October 2019

Revised: March 2020

Published: June 2020

X. He was supported by NSFC Grants (11771468, 11971027, 11271386), and W. Zou by NSFC Grants (11771234, 11926323, 11371212).

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Abstract

In this paper, we study equations involving fully nonlinear nonlocal operators

$ \mathcal {F}_{\alpha}(u(x)) = C_{n, \alpha}P.V.\int_{ \mathbb R^n}\frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}}dz = f(u(x)), \; \; \; x\in \mathbb R^n. $

We shall establish a maximum principle for anti-symmetric functions on any half space, and obtain key ingredients for proving the symmetry and monotonicity for positive solutions to the fully nonlinear nonlocal equations. Especially, a Liouville theorem is derived, which will be useful in carrying out the method of moving planes on unbounded domains for a variety of problems with fully nonlinear nonlocal operators.

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Mathematics Subject Classification: Primary: 35J60; 35J20; Secondary: 35R11; 35S15.

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