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

We study the following nonlinear Schrödinger equation $iu_t=-Δu+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2,$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:= \|Q\|_2^2$, where $Q$ is the unique (up to translations) positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of the standing waves for the above equation as $q\nearrow 2$.

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