Excited states for two-component Bose-Einstein condensates in dimension two

Abstract

We consider the following two-component Bose-Einstein condensates (BEC) by investigating the associated L2-critical problem{−Δu1+V1(x)u1=a1u13+βu1u22+μu1inR2,−Δu2+V2(x)u2=a2u23+βu12u2+μu2inR2, with the constraint ∫R2(u12+u22)dx=1.Ground states and excited states are typically described in BEC phenomenon. In this paper, we prove the existence of the general exited states by use of the finite dimensional reduction method combined with the local Pohozaev identities. Precisely, we construct vector peak solutions as a1+a2−2βa1a2−β2→1ka⁎, where a1,a2>0,β∈(−a1a2,min⁡{a1,a2})∪(max⁡{a1,a2},+∞), k is a positive integer, a⁎=∫R2W2 and W is the unique positive solution of −Δw+w=w3 in R2. Our results generalize those obtained in [9], where Guo-Li-Wei-Zeng proved the existence of the ground states for fixed 0<a1,a2≤a⁎ and β>0 by variational methods.Compared with the single equation, the difficulties in the study of BEC systems come from the interspecies interaction between the components, which requires subtle estimates.