Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations

We study the following nonlinear Schrödinger equationiut=−Δu+V(x)u−a|u|qu,(t,x)∈R1×R2, where a>0a>0, q∈(0,2)q∈(0,2), and V(x)V(x) is some type of trapping potential. For any fixed a>a⁎:=‖Q‖22, where Q   is the unique (up to translations) positive radial solution of Δu−u+u3=0Δu−u+u3=0 in R2R2, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of standing waves for the above equation as q↗2q↗2.