This paper is concerned with the properties of L 2 -normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength a > 0 approaches a critical value a * , each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as a ր a * . However, when a > 0 is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.

We are interested in the attractive Gross-Pitaevskii (GP) equation in R 2 , where the external potential V (x) vanishes on m disjoint bounded domains Ω i ⊂ R 2 (i = 1, 2, · · · , m) and V (x) → ∞ as |x| → ∞, that is, the union of these Ω i is the bottom of the potential well. By making some delicate estimates on the energy functional of the GP equation, we prove that when the interaction strength a approaches some critical value a * the ground states concentrate and blow up at the center of the incircle of some Ω j which has the largest inradius. Moreover, under some further conditions on V (x) we show that the ground states of GP equations are unique and radially symmetric at leat for almost every a ∈ (0, a * ).

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In this paper, we consider the following Schrödinger-Poisson systemwhere λ, μ are positive parameters, p ∈ (1, 5), g(x) ∈ L ∞ (R 3 ) is nonnegative and g(x) ≡ 0 on a bounded domain in R 3 . In this case, μg(x) represents a potential well that steepens as μ getting large. If μ = 0, (P λ ) was well studied in Ruiz (2006) [18]. If μ = 0 and g(x) is not radially symmetric, it is unknown whether (P λ ) has a nontrivial solution for p ∈ (1, 2). By priori estimates and approximation methods we prove that (P λ ) with p ∈ (1, 2) has a ground state if μ large and λ small. In the meantime, we prove also that (P λ ) with p ∈ [3, 5) has a nontrivial solution for any λ > 0 and μ large. Moreover, some behaviors of the solutions of (P λ ) as λ → 0, μ → +∞ and |x| → +∞ are discussed.