Normalized solutions for a kind of mass–supercritical Schrödinger–Choquard equations

We investigate the normalized solutions for the following Schrödinger–Choquard equations constrained to $L^2$-sphere with general potential:

$$\begin{array}{r}\{\begin{array}{ll}-\mathrm{\Delta }v-V(x)v-\lambda v=(I_{\alpha }\ast F(v))f(v),& \mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ {\int }_{{\mathbb{R}}^N}|v{|}^2\mathrm{d}x=a,& \mathrm{i}\mathrm{n}{\mathbb{R}}^N,\end{array}\end{array}$$

where $N\ge 3$, $a>0$ is a prescribed mass, $V\in C^1({\mathbb{R}}^N,\mathbb{R})$ is an external potential, $f\in C(\mathbb{R},\mathbb{R})$, $I_{\alpha }:{\mathbb{R}}^N\to \mathbb{R}$ denotes the Riesz potential with order $\alpha \in (0,N)$, and $\lambda \in \mathbb{R}$ is not prescribed in advance but appears as a Lagrange multiplier. By using critical point theory, we develop reliable arguments to construct the existence results of normalized solutions to this kind of Schrödinger–Choquard equations. The obtained results generalize and improve existing results in the literature.