Abstract
We study the collapse of a many-body system which is used to model two-component Bose-Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for $N$ identical bosons in $\mathbb R^2$, interacting with potentials rescaled in the mean-field manner $-N^{2\beta-1}w^{(\sigma)}(N^{\beta}x)$ with $\int_{\mathbb R^{2}}w^{(\sigma)}(x){\rm d}x=1$. Assuming that $0<\beta<1/2$, we first show that the leading order of the quantum energy is captured correctly by the Gross-Pitaevskii energy. Secondly, we investigate the blow-up behavior of the quantum energy as well as the ground states when $N\to\infty$ and either the total interaction strength of intra-species and inter-species or the strengths of intra-species interactions of each component approaches sufficiently slowly a critical value, which is the critical strength for the focusing Gross-Pitaevskii functional. We prove that the many-body ground states fully condensate on the (unique) Gagliardo-Nirenberg solution.
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