  In this paper, we consider the following nonlinear Schr\"{o}dinger equations
with mixed nonlinearities: \begin{eqnarray*} \left\{\aligned &-\Delta u=\lambda
u+\mu |u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ &u\in
H^1(\bbr^N),\quad\int_{\bbr^N}u^2=a^2, \endaligned\right. \end{eqnarray*} where
$N\geq3$, $\mu>0$, $\lambda\in\mathbb{R}$ and $2<q<2^*$. We prove in this paper
\begin{enumerate} \item[$(1)$]\quad Existence of solutions of mountain-pass
type for $N=3$ and $2<q<2+\frac{4}{N} $. \item[$(2)$]\quad Existence and
nonexistence of ground states for $2+\frac{4}{N}\leq q<2^*$ with $\mu>0$ large.
\item[$(3)$]\quad Precisely asymptotic behaviors of ground states and
mountain-pass solutions as $\mu\to0$ and $\mu$ goes to its upper bound.
\end{enumerate} Our studies answer some questions proposed by Soave in
\cite[Remarks~1.1, 1.2 and 8.1]{S20}.
