Existence of normalized solutions for a Sobolev supercritical Schrödinger equation

This paper studies the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth:

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

where $p>2^{\ast }:=\frac{2N}{N-2}$, $N\ge 3$, $a>0$, $\lambda \in \mathbb{R}$ is an unknown Lagrange multiplier, $V\in C({\mathbb{R}}^N,\mathbb{R})$, $f$ satisfies weak mass subcritical conditions. By employing the truncation technique, we establish the existence of normalized solutions to this Sobolev supercritical problem. Our primary contribution lies in our initial exploration of the case $p>2^{\ast }$, which represents an unfixed frequency problem.