  We are interested in the existence and asymptotical behavior for the least
energy solutions of the following fractional eigenvalue problem
  \begin{equation*}
  (P)\quad (-\Delta)^{s}u+V(x)u=\mu u+am(x)|u|^{\frac{4s}{N}}u,\quad
\int_{\mathbb{R}^{N}}|u|^{2}dx=1,\ u\in H^{s}(\mathbb{R}^{N}),
  \end{equation*}
  where $s\in(0,1)$, $\mu\in\mathbb{R}$, $a>0$, $V(x)$ and $m(x)$ are
$L^{\infty}(\mathbb{R}^{N})$ functions with $N\geq2$. We prove that there is a
threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution
$u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow
a_s^{*}$, at some point $x_0 \in \mathbb{R}^N$, which makes $V(x_0)$ be the
minimum and $m(x_0)$ be the maximum. Moreover, the precise blowup rates for
$u_a$ are obtained under suitable conditions on $V(x)$ and $m(x)$.
