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

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 \{ {array}{l} -Δ u + u^3+β uv^2=λ u,\\ -Δ v + v^3+β u^2v=μ v,\\ u,v∈ H^1_0 (Ω), u,v>0,\\ {array} . as the interspecies scattering length β goes to +∞ . For this system we consider the associated energy functionals J_β , β∈(0,+∞) , with L^2 -mass constraints, which limit J_∞ (as β→+∞ ) 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.