Multiplicity of normalized solutions for a Schr\"{o}dinger equation with critical growth in ${\mathbb{R}}^N$

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr"{o}dinger equation with critical growth \begin{align*} \left{ \begin{aligned} &-\Delta u=\lambda u+\mu |u|^{q-2}u+f(u), \quad \quad \hbox{in }$\mathbb{R}$^N,\ &\int_{$\mathbb{R}$^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a,\mu >0$, $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $q\in (2,2+\frac{4}{N})$ and $f$ has an exponential critical growth when $N=2$, and $f(u)=\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u$ when $N\ge 3$ and $2^{\ast }=\frac{2N}{N-2}$.