Segregated vector solutions for a class of Bose–Einstein systems

Abstract

This paper is concerned with the existence of segregated vector solutions for{−ε2Δui+Pi(x)ui=βiui3+∑j≠inβijuiuj2,x∈Ω,ui>0,x∈Ω,i=1,…,n, where Ω is a bounded or unbounded domain in RN with N=1,2,3, ε>0 is a small parameter, n>1 is an integer, Pi (i=1,…,n) is a potential function, βi>0 (i=1,…,n) is constant and βij=βji>0 (j≠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 βij>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 ε→0+, but the distance between them divided by ε will go to infinity.