Abstract
We consider the nonlinear elliptic system
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 β , v β ) such that u β − v β changes sign precisely k times in the radial variable. Furthermore, as \(\beta \to -\infty\), after passing to a subsequence, u β → w + and v β → w − uniformly in \(\mathbb B\), where w = w +− w − has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δw − 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.
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Wei, J., Weth, T. Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations. Arch Rational Mech Anal 190, 83–106 (2008). https://doi.org/10.1007/s00205-008-0121-9
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DOI: https://doi.org/10.1007/s00205-008-0121-9