On the collapse and concentration of Bose–Einstein condensates with inhomogeneous attractive interactions

Abstract

We consider two-dimensional Bose-Einstein condensates with inhomogeneous attractive interactions 0 < m(x) <= 1, which can be described by the Gross-Pitaevskii functional. We prove that minimizers exist if and only if the interaction strength satisfies a < a* = parallel to Q parallel to(2)(2), where Q is the unique positive radial solution of Delta u - u + u = 0 in R-2. The concentration behavior and symmetry breaking of minimizers as a approaches a* are also analyzed, where all the mass concentrates at a global minimum point x(0) of the trapping potential V(x), provided that x(0) is also a global maximum point of m(x).