Multiplicity of Normalized Solutions to a Class of Non-autonomous Choquard Equations

In this paper, we consider the multiplicity of solutions to the following Choquard equation \(\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=\lambda u+\varepsilon ^{-\alpha }(I_\alpha *[h( x)|u|^p])h(x)|u|^{p-2}u\quad \text{ in } {\mathbb {R}}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill -{\epsilon }^2\mathrm{\Delta }u+V(x)u=\lambda u+{\epsilon }^{-\alpha }(I_{\alpha }{\ast [h(x)|u|}^p{])h(x)|u|}^{p-2}u\text{in}{\mathbb{R}}^N,\end{array}$$ with a prescribed mass \(\begin{aligned} \int _{{\mathbb {R}}^N}|u|^2dx=a\varepsilon ^N, \end{aligned}\) $$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{|u|}^{2}dx=a{\epsilon }^N,\end{array}$$ where \(N\ge 1\) $N\ge 1$ , \(a,\varepsilon >0\) $a,\epsilon >0$ , \(\alpha \in (0,N)\) $\alpha \in (0,N)$ , \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\) $\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}$ , \(I_\alpha \) $I_{\alpha }$ is the Riesz potential, \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier, \(h: {\mathbb {R}}^{N} \rightarrow [0,\infty )\) $h:{\mathbb{R}}^N\to [0,\infty )$ is a bounded and continuous function and the potential V is a continuous function. Under some assumptions on V, we show that when \(\varepsilon \) $\epsilon$ is small enough the numbers of normalized ground states are at least the numbers of global maximum points of h.