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

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

as the interspecies scattering length $\beta$ goes to $+\infty$. For this system we consider the associated energy functionals $J_{\beta }$, $\beta \in (0,+\infty )$, with $L^2$-mass constraints, which limit $J_{\infty }$ (as $\beta \to +\infty$) 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.