Monotonicity and radial symmetry results for Schrödinger systems with fractional diffusion

Jing Li

Vol. 294 (2018), No. 1, 107–121

DOI: 10.2140/pjm.2018.294.107

Abstract

We consider a nonlinear Schrödinger system with fractional diffusion

$$\left\{\begin{array}{cc}{\left(-\Delta \right)}^{\alpha ∕2}u\left(x\right)+A\left(x\right)u\left(x\right)=v^p\left(x\right)\hfill & \text{ in }\Omega ,\hfill \\ {\left(-\Delta \right)}^{\beta ∕2}v\left(x\right)+B\left(x\right)v\left(x\right)=u^q\left(x\right)\hfill & \text{ in }\Omega ,\hfill \\ u\left(x\right)\ge 0,v\left(x\right)\ge 0\hfill & \text{ in }\Omega ,\hfill \\ u\left(x\right)=v\left(x\right)=0\hfill & \text{ on }{\Omega }^C,\hfill \end{array}\right.$$

where $\Omega$ is an unbounded parabolic domain. We first establish a narrow region principle. Using this principle and a direct method of moving planes, we obtain the monotonicity of nonnegative solutions and the Liouville-type result for the nonlinear Schrödinger system with fractional diffusion. We also obtain the radially symmetric result of positive solutions for the system in the unit ball when $A\left(x\right)$ and $B\left(x\right)$ are constants.

Keywords

fractional Schrödinger system, narrow region principle, direct method of moving planes, monotonicity, radially symmetric

Mathematical Subject Classification 2010

Primary: 35J60

Milestones

Received: 28 December 2016

Revised: 1 June 2017

Accepted: 11 September 2017

Published: 5 January 2018