On the existence of multiple normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$(-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N,$$ subject to the constraint $$\int_{\mathbb{R}^N}|u|^2 \mathrm{d}x=a>0,$$ where $N>2 s, s \in(0,1), \alpha \in(0, N), \frac{N+\alpha}{N}<q<\frac{N+2 s+\alpha}{N}<p\leq \frac{N+\alpha}{N-2 s}$, $\epsilon>0$ is a parameter, and $\lambda \in \mathbb{R}$ serves as an unknown parameter acting as a Lagrange multiplier. By employing the Lusternik-Schnirelmann category theory, we estimate the number of normalized solutions to this problem by virtue of the category of the set of minimum points of the potential function $V$.