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Radial solutions and phase separation in a system of two coupled Schrödinger equations. (English) Zbl 1161.35051
Summary: We consider the nonlinear elliptic system \[ \begin{cases} -\Delta u +u - u^3 -\beta v^2u = 0\quad&\text{in }\mathbb B, \\ -\Delta v +v - v^3 -\beta u^2v = 0\quad& \text{in } \mathbb B,\\ u,v > 0 \quad \text{in } \mathbb B,\quad u=v=0 \quad& \text{on } \partial \mathbb B, \end{cases} \] where \(N\leqq 3\) and \(\mathbb B \subset \mathbb {R}^N\) is the unit ball. We show that, for every \(\beta \leqq -1\) and \(k \in \mathbb N\), the above problem admits a radially symmetric solution \((u_\beta,v_\beta)\) such that \(u_\beta - v_\beta\) changes sign precisely \(k\) times in the radial variable. Furthermore, as \(\beta \to -\infty\), after passing to a subsequence, \(u_\beta \rightarrow w ^+\) and \(v _\beta \rightarrow w^{-}\) uniformly in \(\mathbb B\), where \(w = w^{+} - w^{-}\) has precisely \(k\) nodal domains and is a radially symmetric solution of the scalar equation \(\Delta w - w + w^3 = 0\) in \(\mathbb B\), \(w = 0\) on \(\partial \mathbb B\). Within a Hartree-Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose-Einstein double condensates with strong repulsion.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
81V10 Electromagnetic interaction; quantum electrodynamics
35B40 Asymptotic behavior of solutions to PDEs
35A35 Theoretical approximation in context of PDEs
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