, Volume 190, Issue 1, pp 83-106
Date: 19 Mar 2008

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

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Abstract

We consider the nonlinear elliptic system $$\left \{ \begin{aligned} -&\Delta u +u - u^3 -\beta v^2u = 0\quad \rm{in}\, \mathbb B,\\ -&\Delta v +v - v^3 -\beta u^2v = 0\quad \rm{in}\, \mathbb B,\\ &u,v > 0 \quad \rm{in}\, \mathbb B,\quad u=v=0 \quad \rm{on}\, \partial \mathbb B, \end{aligned} \right.$$ 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 Δww + 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.

Communicated by C.A. Stuart