Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials

In this paper we consider a two-dimensional attractive Bose-Einstein condensate with periodic potential, described by Gross-Pitaevskii (GP) functional. By concentration-compactness lemma we show that minimizers of this functional exist when the interaction strength a satisfies a⁎0, and there is no minimizer for a≥a⁎. When a approaches a⁎, using concentration-compactness arguments again we obtain an optimal energy estimate depending on the shape of periodic potential. Moreover, we analyze the mass concentration.