New vector solutions for the cubic nonlinear schrödinger system

Abstract

In this paper, we construct a family of new solutions for the following nonlinear Schrödinger system:

$$\left\{{\matrix{{- \Delta u + P(y)u = \mu {u^3} + \beta u{\upsilon ^2},} & {u > 0,\,\,{\rm{in}}\,{\mathbb{R}^3},} \cr {- \Delta \upsilon + Q(y)\upsilon = v{\upsilon ^3} + \beta {u^2}\upsilon ,} & {\upsilon > 0,\,\,{\rm{in}}\,{\mathbb{R}^3},} \cr}} \right.$$

where P(y), Q(y) are positive radial potentials, μ > 0, v > 0 and \(\beta \in \mathbb{R}\). Motivated by the doubling construction of the entire finite energy sign-changing solution for the Yamabe equation in M. Medina and M. Musso (J. Math. Pures Appl. 2021), by using another type of building blocks, which are not equal to the ones adopted in S. Peng and Z.-Q. Wang (Arch. Ration. Mech. Anal. 2013), we successfully construct new segregated and synchronized vector solutions for the nonlinear Schrödinger system with more complex concentration structure. Our results extend the main results of S. Peng and Z.-Q. Wang (Arch. Ration. Mech. Anal. 2013), and in particular, for the segregated case, we well complement the previous works when the potentials P(y) and Q(y) decay in different rates.