A BEC System with Dimensions $$N = 2, 3$$: Ground State Solutions

Abstract

As introduced in Chap. 1, we study the ground state solutions of system ( 1.2) in the entire space \(\mathbb R^N\) with \(N=2, 3\). Precisely, motivated by Sirakov’s previous work, we prove some uniqueness results of positive (ground state) solutions for the special case \(\lambda _1=\lambda _2\). These give partial answers to Sirakov’s conjecture. For the general case \(\lambda _1\ne \lambda _2\), we prove a sharp result on the parameter range for the existence of ground state solutions. The asymptotic behaviors of ground state solutions can be investigated as a corollary. We also prove a nonexistence result about positive solutions. These results answer partially some open questions raised by Ambrosetti, Colorado and Sirakov. Our proof is mainly applying asymptotic analysis together with the classical bifurcation theory.