Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in R2

We investigate the ground states of 3-component Bose-Einstein condensates with harmonic-like trapping potentials in R-2, where the intra-component interactions mu(i) and the inter- component interactions beta(ij) = beta(ji) (i, j = 1, 2, 3, i not equal j) are all attractive. We display the regions of mu(i) and beta(ij) for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as beta ij NE arrow beta(*)(ij) := a* + 1/2 root(a* - mu(i))(a* - mu(j)), where 0 < mu(i) < a* := parallel to w parallel to(2)(2) are fixed and w is the unique positive solution of Delta w - w + w(3) = 0 in H1(R2). The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.