Archive for Rational Mechanics and Analysis

, Volume 190, Issue 1, pp 83–106

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

Article

DOI: 10.1007/s00205-008-0121-9

Cite this article as:
Wei, J. & Weth, T. Arch Rational Mech Anal (2008) 190: 83. doi:10.1007/s00205-008-0121-9

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 (uβ, vβ) such that uβvβ changes sign precisely k times in the radial variable. Furthermore, as \(\beta \to -\infty\), after passing to a subsequence, uβw+ and vβw uniformly in \(\mathbb B\), where w = w+w has precisely k nodal domains and is a radially symmetric solution of the scalar equation Δww + w3 = 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.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsChinese University of Hong KongShatinHong Kong
  2. 2.Mathematische InstitutUniversitat GiessenGiessenGermany

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