Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

Abstract

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in R2, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in R2, and solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. L2-norm constaints). Under a certain type of trapping potentials Vi(x) (i = 1,2), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of Vi(x)) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.