Multiple normalized solutions to Schrödinger equations in \(\mathbb {R}^N\) with critical growth and potential

In this paper, we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equations with critical growth and potential \(\begin{aligned} \left\{ \begin{array}{lll} -\Delta u+V(\epsilon x)u =\lambda u+\mu |u|^{q-2}u+f(u) \quad \text {in } \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^2\textrm{d}x=a, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+V(ϵx)u=\lambda u+{\mu |u|}^{q-2}u+f(u)\text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^2\text{d}x=a,\hfill \end{array}\right)\end{array}$$ where \( a>0, \mu>0, q\in (2,2+4/N), \epsilon >0\) $a>0,\mu >0,q\in (2,2+4/N),ϵ>0$ is a small parameter, \( \lambda \in \mathbb {R} \) $\lambda \in \mathbb{R}$ will arise as Lagrange multiplier, the potential \(V:\mathbb {R}^N\rightarrow \left[ 0,+\infty \right) \) $V:{\mathbb{R}}^N\to \left[0,,,+,\infty \right)$ is a continuous function that satisfies some suitable conditions and \(f\in C(\mathbb {R},\mathbb {R}) \) $f\in C(\mathbb{R},\mathbb{R})$ enjoys critical exponential growth of Trudinger–Moser type when \(N=2\) $N=2$ , and \(f(u)=|u|^{2^*-2}u\) $f(u)={|u|}^{2^{\ast }-2}u$ when \(N\ge 3\) $N\ge 3$ and \(2^*=\frac{2N}{N-2}\) $2^{\ast }=\frac{2N}{N-2}$ . When \( \epsilon >0 \) $ϵ>0$ is sufficiently small, we prove that the existence of normalized solutions by applying truncation techniques, and the relationship between the number of positive solutions and the topology of the set where V attains its minimum is obtained by using Ljusternik–Schnirelmann theory.