In this paper, we consider the multiplicity and asymptotics of standing waves for the energy critical half-wave, which are solutions to (0.1) \({{\sqrt{-\Delta} u}}=\lambda u+\mu{\vert u \vert}^{q-2}u+{\vert u \vert}^{2^{\ast}-2}u, \quad u \in H^{1/2}({\mathbb R}^{N}),\) under the constraint \(\int_{{\mathbb R}^{N}} u^{2}=a^{2}\) , where N ≥ 2, a > 0, \(q \in (2,2+{2 \over N}), 2^{\ast}={{2N} \over {N-1}}\) and λ ∈ ℝ appears as a Lagrange multiplier. We show that (0.1) admits a ground state ua and an excited state va, which are characterized by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Several asymptotic properties of {ua}, {va} are obtained and it is worth pointing out that we get a precise description of {ua} as a → 0+ without needing any uniqueness condition on the related limit problem. Finally, assuming local well-posedness, we prove that the set of ground states is stable under the half-wave evolution.
