In this paper the authors continue their analysis of two-component attractive Bose-Einstein condensates by studying new pattern formations of ground states \((u_1,u_2)\). It is assumed the trapping potentials are homogeneous in \(\mathbb{R}^2\), the intraspecies interaction \((-a,-b)\) and the interspecies interaction \(-\beta\) are both attractive, i.e., \(a,b,\beta>0\).
The particular Gross-Pitaevskii system considered is \[ \begin{cases} -\Delta u_1+V_1(x)u_1=\mu u_1+au_1^3+\beta u_2^2u_1 \ \text{ in } \mathbb{R}^2,\\ -\Delta u_2+V_2(x)u_2=\mu u_2+bu_2^3+\beta u_1^2u_2 \ \text{ in } \mathbb{R}^2. \end{cases} \] Ground states of such systems can be described equivalently by minimizers of the \(L^2\)-critical constraint variation problem \[ e(a,b,\beta)=\inf_{(u_1,u_2)\in \chi \ : \int_{\mathbb{R}^2}u_1^2+u_2^2=1}E_{a,b,\beta}(u_1,u_2), \quad a, b, \beta>0, \] where the Gross-Pitaevskii energy function is given by \[ E_{a,b,\beta}(u_1,u_2) =\int_{\mathbb{R}^2}\left(|\nabla u_1|^2+|\nabla u_2|^2\right)+\int_{\mathbb{R}^2}\left(V_1(x)u_1^2+V_2(x)u_2^2\right) \] \[ -\int_{\mathbb{R}^2}\left(\frac{a}{2}u_1^4+\frac{b}{2}u_2^4+\beta u_1^2u_2^2\right), \] and the space of allowable functions is \(\chi=\mathcal{H}_1(\mathbb{R}^2)\times\mathcal{H}_2(\mathbb{R}^2)\), with \[ \mathcal{H}_i(\mathbb{R}^2)=\left\{u\in H^1(\mathbb{R}^2) \ : \ \int_{\mathbb{R}^2}V_i(x)u^2<\infty\right\}. \] The trapping potentials \(V_1,V_2\) are assumed to be homogeneous.
The main goal of this paper is to investigate new pattern formations of nonnegative minimizers \((u_1,u_2)\) for \(e(a,b,\beta)\), under certain regimes for \(a,b,\beta\).
If \(0< b < a^* =||w||_2^2\) and \(0< \beta< a^*\) are fixed, where \(w\) is the unique positive solution of \(\Delta w-w+w^3=0\) in \(\mathbb{R}^2\), the authors obtain semi-trivial behavior of \((u_1,u_2)\) as \(a\nearrow a^*\), that is, \(u_1\) concentrates at a unique point and \(u_2\equiv 0\) in \(\mathbb{R}^2\).
Under a nondegeneracy assumption, if \(0< b< a^*\) and \(a^*\le \beta<\beta^*=a^*+\sqrt{(a^*-a)(a^*-b)}\), the refined spike profile and the uniqueness of \((u_1,u_2)\) as \(a\nearrow a^*\) are analyzed. It is shown that \((u_1,u_2)\) must be unique, \(u_1\) concentrates at a unique point, and \(u_2\) either blows-up or vanishes, depending on how \(\beta\) approaches \(a^*\).