Vol. 299, No. 1, 2019

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Symmetry and nonexistence of solutions for a fully nonlinear nonlocal system

Biran Zhang and Zhongxue Lü

Vol. 299 (2019), No. 1, 237–255
Abstract

We study the system involving fully nonlinear nonlocal operators:

Fα(u(x)) = Cn,α PVnG(u(x) u(y)) |x y|n+α dy = f(u(x),v(x)), Fβ(v(x)) = Cn,β PVnG(v(x) v(y)) |x y|n+β dy = g(u(x),v(x)).

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

Fα(u(x)) = up(x) + vq(x),F β(v(x)) = vp(x) + uq(x)

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
Authors
Biran Zhang
School of Mathematics and Statistics
Jiangsu Normal University
Xu Zhou
China
Zhongxue Lü
School of Mathematics and Statistics
Jiangsu Normal University
Xuzhou
China