Existence of normalized solutions for the Schrödinger equation

In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.

$$\begin{array}{rl}& \{\begin{array}{ll}-\mathrm{\Delta }u=\lambda u+\mu |u{|}^{q-2}u+|u{|}^{p-2}u& \text{in}{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=a^2,\end{array}\end{array}$$

where $N⩾3$, $2<q<2+\frac{4}{N}$, $p=2^{\ast }=\frac{2N}{N-2}$, $a,\mu >0$ and $\lambda \in \mathbb{R}$ is a Lagrange multiplier. Since the existence result for $2+\frac{4}{N}<p<2^{\ast }$ has been proved, using an approximation method, that is let $p\to 2^{\ast }$, we obtain that there exists a mountain-pass type solution for $p=2^{\ast }$.