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: $$(-\mathrm{\Delta }{)}^{s}u+V(\epsilon x)u=\lambda u+(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^q)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,x\in {\mathbb{R}}^N,$$ subject to the constraint $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{d}x=a>0,$$ where $N>2s,s\in (0,1),\alpha \in (0,N),\frac{N+\alpha }{N}<q<\frac{N+2s+\alpha }{N}<p\le \frac{N+\alpha }{N-2s}$, $\epsilon >0$ 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 $V$.