A direct method of moving planes for fully nonlinear nonlocal operators and applications

Yuxia Guo; Shaolong Peng

doi:10.3934/dcdss.2020462

Article Contents

2021, Volume 14, Issue 6: 1871-1897. Doi: 10.3934/dcdss.2020462

This issue Previous Article Structure of positive solutions to a class of Schrödinger systems Next Article Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space

A direct method of moving planes for fully nonlinear nonlocal operators and applications

Yuxia Guo^1, , and
Shaolong Peng^1,

Department of Mathematics, Tsinghua University, Beijing, 100084, China

^* Corresponding author: Yuxia Guo
^* Corresponding author: Yuxia Guo

Received: June 30, 2020

Early access: November 2020

Published: June 2021

The first author is supported by NSFC (No. 11771235) and the second author supported by NSFC (No. 11971049)

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Abstract

In this paper, we are concerned with the following generalized fully nonlinear nonlocal operators:

$ F_{s,m}(u(x)) = c_{N,s} m^{ \frac{N}{2}+s} P.V. \int_{\mathbb{R}^{N}} \frac{G(u(x)-u(y))}{|x-y|^{ \frac{N}{2}+s}} K_{ \frac{N}{2}+s}(m|x-y|)dy+m^{2s}u(x), $

where $ s\in (0,1) $ and mass $ m>0 $. By establishing various maximal principle and using the direct method of moving plane, we prove the monotonicity, symmetry and uniqueness for solutions to fully nonlinear nonlocal equation in unit ball, $ \mathbb{R}^{N} $, $ \mathbb{R}^{N}_{+} $ and a coercive epigraph domain $ \Omega $ in $ \mathbb{R}^N $ respectively.

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Mathematics Subject Classification: Primary:35R11;Secondary:35B06, 35B53.

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