Symmetry and nonexistence of solutions for a fully nonlinear nonlocal system

Biran Zhang and Zhongxue Lü

Vol. 299 (2019), No. 1, 237–255

DOI: 10.2140/pjm.2019.299.237

Abstract

We study the system involving fully nonlinear nonlocal operators:

$$\begin{array}{cc}\hfill F_{\alpha }\left(u\left(x\right)\right)& =C_{n,\alpha }PV<!--nolimits-->{\int \; <!--nolimits-->}_{{ℝ}^n}\frac{G\left(u\left(x\right)-u\left(y\right)\right)}{|x-y{|}^{n+\alpha }}dy=f\left(u\left(x\right),v\left(x\right)\right),\hfill \\ \hfill F_{\beta }\left(v\left(x\right)\right)& =C_{n,\beta }PV<!--nolimits-->{\int \; <!--nolimits-->}_{{ℝ}^n}\frac{G\left(v\left(x\right)-v\left(y\right)\right)}{|x-y{|}^{n+\beta }}dy=g\left(u\left(x\right),v\left(x\right)\right).\hfill \end{array}$$

We will prove the symmetry and monotonicity for positive solutions to the nonlinear system in whole space by using the method of moving planes. To achieve it, a narrow region principle and a decay at infinity are established. Further more, nonexistence of positive solutions to the nonlinear system on a half space is derived. In addition, the symmetry and monotonicity in whole space for positive solutions to a fully nonlinear nonlocal system

$$F_{\alpha }\left(u\left(x\right)\right)=-u^p\left(x\right)+v^q\left(x\right),F_{\beta }\left(v\left(x\right)\right)=-v^p\left(x\right)+u^q\left(x\right)$$

can be derived.

Keywords

fully nonlinear nonlocal operator, narrow region principle, decay at infinity, method of moving planes

Mathematical Subject Classification 2010

Primary: 35B06, 35B09, 35B50, 35B53

Milestones

Received: 22 December 2017

Revised: 16 May 2018

Accepted: 24 June 2018

Published: 18 April 2019