Vol. 294, No. 1, 2018

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Monotonicity and radial symmetry results for Schrödinger systems with fractional diffusion

Jing Li

Vol. 294 (2018), No. 1, 107–121

We consider a nonlinear Schrödinger system with fractional diffusion

(Δ)α2u(x) + A(x)u(x) = vp(x) in Ω, (Δ)β2v(x) + B(x)v(x) = uq(x)  in Ω, u(x) 0,v(x) 0  in Ω, u(x) = v(x) = 0  on ΩC,

where Ω 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(x) and B(x) are constants.

fractional Schrödinger system, narrow region principle, direct method of moving planes, monotonicity, radially symmetric
Mathematical Subject Classification 2010
Primary: 35J60
Received: 28 December 2016
Revised: 1 June 2017
Accepted: 11 September 2017
Published: 5 January 2018
Jing Li
College of Physics and Materials Science
College of Mathematics and Information Science
Henan Normal University