JournalsjemsVol. 14, No. 4pp. 1245–1273

Convergence of minimax structures and continuation of critical points for singularly perturbed systems

  • Susanna Terracini

    Università di Torino, Italy
  • Gianmaria Verzini

    Politecnico di Milano, Italy
  • Benedetta Noris

    Università degli Studi di Milano-Bicocca, Italy
  • Hugo Tavares

    IST - Universidade de Lisboa, Portugal
Convergence of minimax structures and continuation of critical points for singularly perturbed systems cover
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Abstract

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system

{Δu+u3+βuv2=λu,Δv+v3+βu2v=μv,u,vH01(Ω),u,v>0,\left\{ \begin{array}{l} -\Delta u + u^3+\beta uv^2=\lambda u,\\ -\Delta v + v^3+\beta u^2v=\mu v,\\ u,v\in H^1_0 (\Omega),\quad u,v>0,\\ \end{array} \right.

as the interspecies scattering length β\beta goes to ++\infty. For this system we consider the associated energy functionals JβJ_\beta, β(0,+)\beta\in(0,+\infty), with L2L^2-mass constraints, which limit JJ_\infty (as β+\beta\to+\infty) is strongly irregular. For such functionals, we construct multiple critical points via a common minimax structure, and prove convergence of critical levels and optimal sets. Moreover we study the asymptotics of the critical points.

Cite this article

Susanna Terracini, Gianmaria Verzini, Benedetta Noris, Hugo Tavares, Convergence of minimax structures and continuation of critical points for singularly perturbed systems. J. Eur. Math. Soc. 14 (2012), no. 4, pp. 1245–1273

DOI 10.4171/JEMS/332