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.