Segregated vector solutions for a class of Bose–Einstein systems

Abstract

This paper is concerned with the existence of segregated vector solutions for

$$\{\begin{array}{c}-{\epsilon }^2\mathrm{\Delta }{u}_i+P_i(x)u_i={\beta }_i{u}_i^3+\sum _{j\ne i}^n{\beta }_{ij}{u}_i{u}_j^2,x\in \Omega ,\hfill \\ u_i>0,x\in \Omega ,i=1,\dots ,n,\hfill \end{array}$$

where Ω is a bounded or unbounded domain in ${\mathbb{R}}^N$ with $N=1,2,3$, $\epsilon >0$ is a small parameter, $n>1$ is an integer, $P_i$ ($i=1,\dots ,n$) is a potential function, ${\beta }_i>0$ ($i=1,\dots ,n$) is constant and ${\beta }_{ij}={\beta }_{ji}>0$ ($j\ne i$) is coupling constant. This system describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive in optics and Bose–Einstein condensates. For ${\beta }_{ij}>0$, which corresponds to the synchronization case for the above system with constant potentials, we prove that the system has multiple positive vector solutions, whose components may have spikes clustering at the same point as $\epsilon \to 0^{+}$, but the distance between them divided by ε will go to infinity.