Segregated vector solutions for nonlinear schrödinger systems in ℝ2

We study the following nonlinear Schrödinger system {−Δu+P(|x|)u=μu3+βv2u, x∈ℝ2,−Δv+Q(|x|)v=vu3+βu2v, x∈ℝ2,where P(r) and Q(r)   are positive radial functions, μ>0,v>0, and β∈ℝμ>0,v>0, and β∈ℝ is a coupling constant. This type of system arises, particularly, in models in Bose-Einstein condensates theory. Applying a finite reduction method, we construct an unbounded sequence of non-radial positive vector solutions of segregated type when β is in some suitable interval, which gives an answer to an interesting problem raised by Peng and Wang in Remark 4.1 (Arch. Ration. Mech. Anal., 208(2013), 305–339).