Blow-up behavior of ground states for a nonlinear Schr\"{o}dinger system with attractive and repulsive interactions

TL;DR

This paper studies the conditions under which ground states exist for a two-component nonlinear Schrödinger system modeling Bose-Einstein condensates, revealing a threshold for attractive interactions and analyzing the behavior near this critical point.

Contribution

It establishes a precise threshold for the existence of ground states based on interaction strengths and describes the asymptotic concentration behavior of solutions near this threshold.

Findings

Ground states exist if and only if the attractive interaction strength is below a critical value.

No minimizer exists if the interaction exceeds the critical threshold.

Solutions concentrate at the minimum of the trapping potential as parameters approach the threshold.

Abstract

We consider a nonlinear Schr\"odinger system arising in a two-component Bose-Einstein condensate (BEC) with attractive intraspecies interactions and repulsive interspecies interactions in $\mathbb{R}^2$. We get ground states of this system by solving a constrained minimization problem. For some kinds of trapping potentials, we prove that the minimization problem has a minimizer if and only if the attractive interaction strength $a_i (i=1,2)$ of each component of the BEC system is strictly less than a threshold $a^*$. %attractive intraspecies interactions satisfies $a_i< %a^*= \|Q\|_2^2,\ i=1,\,2$, where $Q$ is the unique positive radial solution of $\Delta u-u+u^3=0$ in $\mathbb{R}^2$; in contrast, there is no minimizer if either $a_i > a^*$ for $i=1$ or $2$, or $a_1=a_2=a^*$. Furthermore, as $(a_1, a_2)\nearrow (a^*, a^*)$, the asymptotical behavior for the minimizers of the minimization problem is discussed. Our results show that each component of the BEC system concentrates at a global minimum of the associated trapping potential.