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

Abstract

In this paper, we study the asymptotic properties of solutions for the constrained minimization problems.

$$\begin{aligned} d_{b_p}(p):=\inf _{\{u\in H^s_V({\mathbb {R}}^2): \int _{{\mathbb {R}}^2}|u|^2dx=1\}}I_{p,b_p}(u), \end{aligned}$$

where $s\in (\frac{1}{2},1),$ $p\in (0, 2s)$, $b_p>0$ and

$$\begin{aligned} I_{p,b_p}(u){:=}\frac{1}{2}\int _{{\mathbb {R}}^2}\left( |(-\Delta )^{\frac{s}{2}}u|^2{+}V(x)|u|^2\right) dx{-}\frac{b_p}{p+2}\int _{{\mathbb {R}}^2}|u|^{p+2}dx,\quad u\in H^s_V({\mathbb {R}}^2). \end{aligned}$$

First, when $\lim _{p\nearrow 2s}b_p=b<b^*$, the set of minimizers of $d_{b_p}(p)$ is compact in a suitable space as $p\nearrow 2s$. In addition, when $\lim _{p\nearrow 2s}b_p=b\ge b^*$, by developing suitable trial functions for some fine energy estimates, we prove that all minimizers must blow up and give decay properties of minimizers.