  In this paper, we consider the multiplicity and asymptotics of standing waves
with prescribed mass $\int_{{\mathbb{R}^N}} {{u}^2}=a^2$ to the energy critical
half-wave \begin{equation}\label{eqA0.1} \sqrt{-\Delta}u=\lambda u+\mu|u|^{q-2}
u+|u|^{2^*-2}u,\ \ u\in H^{1/2}(\R^N), \end{equation} where $N\!\geq\! 2$,
$a\!>\!0$, $q \!\in\!\big(2,2+\frac{2}{N}\big)$, $2^*\!=\!\frac{2N}{N-1}$ and
$\lambda\!\in\!\R$ appears as a Lagrange multiplier. We show that
\eqref{eqA0.1} admits a ground state $u_a$ and an excited state $v_a$, which
are characterised by a local minimizer and a mountain-pass critical point of
the corresponding energy functional. Several asymptotic properties of
$\{u_a\}$, $\{v_a\}$ are obtained and it is worth pointing out that we get a
precise description of $\{u_a\}$ as $a\!\to\! 0^+$ without needing any
uniqueness condition on the related limit problem. The main contribution of
this paper is to extend the main results in J. Bellazzini et al. [Math. Ann.
371 (2018), 707-740] from energy subcritical to energy critical case.
Furthermore, these results can be extended to the general fractional nonlinear
Schr\"{o}dinger equation with Sobolev critical exponent, which generalize the
work of H. J. Luo-Z. T. Zhang [Calc. Var. Partial Differ. Equ. 59 (2020)] from
energy subcritical to energy critical case.

Analysis of PDEs