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

TL;DR

This paper investigates the convergence behavior of solutions to a competitive Bose-Einstein condensation system as competition intensifies, focusing on minimax solutions obtained through Krasnoselskii genus theory, and provides partial positive evidence.

Contribution

It offers new insights into the convergence of minimax critical points in singularly perturbed systems related to Bose-Einstein condensation.

Findings

Solutions tend to converge to scalar equation solutions as competition increases

Partial results support the conjecture of convergence for minimax solutions

The approach uses Krasnoselskii genus theory to identify critical points

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.