Asymptotic Behavior of Normalized Solutions for Fractional $L^2$-Critical Schrödinger Equations with a Spatially Decaying Nonlinearity

This paper is devoted to studying the following fractional $L^2$-critical nonlinear Schr\"odinger equation $$(-\Delta)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{\frac{4s-2b}{N}}u(x) = \mu u(x)\,\ \hbox{in}\,\ \mathbb{R}^N,$$ where $\mu\in\mathbb{R}$, $a>0$, $s\in(\frac{1}{2},1)$, $N>2s$, $00$ such that (1.4) has minimizers for $0a^*$. For the case of $a=a^*$, it gives a fact that the existence and non-existence of minimizers depend strongly on the value of $V(0)$. Especially for $V(0)=0$, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as $a\nearrow a^*$. Applying implicit function theorem, the uniqueness of minimizers is also proved for $a>0$ small enough.