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:

$\begin{array}{cc}\hfill {F}_{\alpha }\left(u\left(x\right)\right)& ={C}_{n,\alpha }PV{\int }_{{ℝ}^{n}}\frac{G\left(u\left(x\right)-u\left(y\right)\right)}{|x-y{|}^{n+\alpha }}\phantom{\rule{0.3em}{0ex}}dy=f\left(u\left(x\right),v\left(x\right)\right),\hfill \\ \hfill {F}_{\beta }\left(v\left(x\right)\right)& ={C}_{n,\beta }PV{\int }_{{ℝ}^{n}}\frac{G\left(v\left(x\right)-v\left(y\right)\right)}{|x-y{|}^{n+\beta }}\phantom{\rule{0.3em}{0ex}}dy=g\left(u\left(x\right),v\left(x\right)\right).\hfill \end{array}$

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

${F}_{\alpha }\left(u\left(x\right)\right)=-{u}^{p}\left(x\right)+{v}^{q}\left(x\right),\phantom{\rule{1em}{0ex}}{F}_{\beta }\left(v\left(x\right)\right)=-{v}^{p}\left(x\right)+{u}^{q}\left(x\right)$

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