Limiting behaviors of constrained minimizers for the mass subcritical fractional NLS equations

In this paper, we study the asymptotic properties of solutions for the constrained minimization problems. \(\begin{aligned} d_{b_p}(p):=\inf _{\{u\in H^s_V({\mathbb {R}}^2): \int _{{\mathbb {R}}^2}|u|^2dx=1\}}I_{p,b_p}(u), \end{aligned}\) $$\begin{array}{c}\hfill d_{b_p}(p):=\underset{\{u\in H_V^s({\mathbb{R}}^2):{\int }_{{\mathbb{R}}^2}|u{|}^{2}dx=1\}}{inf}{I}_{p,b_p}(u),\end{array}$$ where \(s\in (\frac{1}{2},1),\) $s\in (\frac{1}{2},1),$ \(p\in (0, 2s)\) $p\in (0,2s)$ , \(b_p>0\) $b_p>0$ and \(\begin{aligned} I_{p,b_p}(u){:=}\frac{1}{2}\int _{{\mathbb {R}}^2}\left( |(-\Delta )^{\frac{s}{2}}u|^2{+}V(x)|u|^2\right) dx{-}\frac{b_p}{p+2}\int _{{\mathbb {R}}^2}|u|^{p+2}dx,\quad u\in H^s_V({\mathbb {R}}^2). \end{aligned}\) $$\begin{array}{c}\hfill I_{p,b_p}(u):=\frac{1}{2}{\int }_{{\mathbb{R}}^2}\left({|(-\mathrm{\Delta })}^{\frac{s}{2}},{u|}^2,+,V,(x),{|u|}^2\right)dx-\frac{b_p}{p+2}{\int }_{{\mathbb{R}}^2}{|u|}^{p+2}dx,u\in H_V^s({\mathbb{R}}^2).\end{array}$$ First, when \(\lim _{p\nearrow 2s}b_p=b<b^*\) ${lim}_{p\nearrow 2s}{b}_p=b<b^{\ast }$ , the set of minimizers of \(d_{b_p}(p)\) $d_{b_p}(p)$ is compact in a suitable space as \(p\nearrow 2s\) $p\nearrow 2s$ . In addition, when \(\lim _{p\nearrow 2s}b_p=b\ge b^*\) ${lim}_{p\nearrow 2s}{b}_p=b\ge b^{\ast }$ , by developing suitable trial functions for some fine energy estimates, we prove that all minimizers must blow up and give decay properties of minimizers.