Normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^{N}$

Abstract: In this paper we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left{ \begin{aligned} &-\Delta u=\lambda 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 $a>0$, $\lambda\in \mathbb{R}$ and $f$ has an exponential critical growth when $N=2$, and $f(t)=\mu |u|^{q-2}u+|u|^{2^*-2}u$ with $q \in (2+\frac{4}{N},2^*)$, $\mu>0$ and $2^*=\frac{2N}{N-2}$ when $N \geq 3$. Our main results complement some recent results for $N \geq 3$ and it is totally new for $N=2$.