Convergence of minimax structures and continuation of critical points for singularly perturbed systems (Q437818)

Summary: 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 \[ -\Delta u+ u^3+\beta uv^2= \lambda u,\qquad -\Delta v+ v^3+\beta u^2 v=\mu v, \] \[ u,v\in H^1_0(\Omega),\quad u,v> 0, \] 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.