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

Abstract

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