Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations

We study the following minimization problem:

${d_{{a_q}}}(q): = \mathop {\inf }\limits_{\{ \int {_{{\mathbb{R}^2}}|u{|^2}dx = 1} \} } {E_{q,{a_q}}}(u),$

where the functional $E_{q,a_q}(·)$ is given by

${{E}_{q,{{a}_{q}}}}(u):=\int_{{{\mathbb{R}}^{2}}}{(|\nabla u(x){{|}^{2}}+V(x)|u(x){{|}^{2}})}dx-\frac{2{{a}_{q}}}{q+2}\int_{{{\mathbb{R}}^{2}}}{|}u(x){{|}^{q+2}}dx.$

Here $a_q>0, \ q∈(0,2)$ and $V(x)$ is some type of trapping potential. Let $a^*:= \|Q\|_2^2$ , where $Q$ is the unique (up to translations) positive radial solution of $Δ u-u+u^3=0$ in $\mathbb{R}^2$ . We prove that if $\lim_{q\nearrow2}a_q=a<a^*$ , then minimizers of $d_{a_q}(q)$ is compact in a suitable space as $q\nearrow2$ . On the contraty, if $\lim_{q\nearrow2}a_q=a≥q a^*$ , by directly using asymptotic analysis, we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers. These conclusions extend the results of Guo-Zeng-Zhou [Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations, J. Differential Equations. 256, (2014), 2079-2100].