Li-qin TANG, Li WANG, Jun WANG
This paper considers the existence of multiple normalized solutions of the following Schrödinger-Choquard equation \begin{align}\nonumber \Bigg\{ & -\Delta u=\lambda u+k(\varepsilon x)(I_{\alpha}\ast|u|^{q})|u|^{q-2}u+\mu(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, &x\in \mathbb{R}^N,\\ & \int_{\mathbb{R}^N}|u|^2 dx=c^2, &x\in \mathbb{R}^N, \end{align} where $c,\varepsilon,\mu>0,\ N\geq 3,\ \alpha\in(0,N),\ \frac{N+\alpha}{N}<q<1+\frac{\alpha+2}{N}<p\leq\frac{N+\alpha}{N-2},\ \lambda \in \mathbb{R}$ is a Lagrange multiplier which is unknown, $I_{\alpha}$ is the Riesz potential, $k: \mathbb{R}^N \rightarrow[0, \infty)$ is a continuous and positive function. When $\varepsilon$ is small enough, we prove that the numbers of normalized solutions are at least the numbers of global maximum points of $k$ by Ekeland's variational principle and truncated skill.
