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We consider a competitive system of two stationary Gross-Pitaevskii equationsarising in the theory of Bose-Einstein condensation, and the correspondingscalar equation. We address the question: "Is it true that every bounded familyof solutions of the system converges, as the competition parameter goes toinfinity, to a pair which difference solves the scalar equation?". We discussthis question in the case when the solutions to the system are obtained asminimax critical points via (weak) L^2 Krasnoselskii genus theory. Our results,though still partial, give a strong indication of a positive answer.

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