On vector solutions of nonlinear Schrödinger systems with mixed potentials

Abstract

In this paper, we are concerned with the following Schrödinger system$\{\begin{array}{cc}-\Delta u+P\left(x\right)u={\mu }_1{u}^3+{\beta }_{1}u{v}^2+{\beta }_{2}u{w}^2,& x\in R^3,\\ -\Delta v+Q\left(x\right)v={\mu }_2{v}^3+{\beta }_{1}v{u}^2+{\beta }_{3}v{w}^2,& x\in R^3,\\ -\Delta w+\lambda w={\mu }_3{w}^3+{\beta }_{2}w{u}^2+{\beta }_{3}w{v}^2,& x\in R^3,\end{array}$ where $\lambda >0$ is a positive constant, $P\left(x\right),Q\left(x\right)$ are continuous positive radial potentials, ${\mu }_i(i=1,2,3)>0$ and ${\beta }_i(i=1,2,3)\in R$ are coupling constants. We mainly investigate the effect of the potentials and the nonlinear coupling on the structure of solutions. Applying the Lyapunov-Schmidt reduction method, we prove the existence of infinitely many positive solutions and sign-changing solutions to the system whose energy can be arbitrarily large. Specifically, we obtain solutions with some of the components synchronized between them while segregated with the rest of the components. Moreover, we also show the existence of another solutions with all components segregated, one of which concentrates at the origin. Our results present vector solutions to the system with different characters. To our knowledge, it is the first time to study systems with three equations involving mixed potentials.