Normalized solutions for a class of fractional Choquard equations with mixed nonlinearities

In this paper we study the following fractional Choquard equation with mixed nonlinearities: $$\{\begin{array}{l}(-\mathrm{\Delta }{)}^{s}u=\lambda u+\alpha (I_{\mu }\ast \mathrm{\mid }u{\mathrm{\mid }}^q)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+(I_{\mu }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,x\in {\mathbb{R}}^N,\\ {\displaystyle {\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{d}x=c^2>0.}\end{array}$$ Here $N>2s$, $s\in (0,1)$, $\mu \in (0,N)$, and the exponents satisfy $$\frac{2N-\mu }{N}<q<p<\frac{2N-\mu }{N-2s},$$ while $\alpha >0$ is a sufficiently small parameter, $\lambda \in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint, and $I_{\mu }$ denotes the Riesz potential. We establish existence and multiplicity results for normalized solutions and, in addition, prove the existence of ground state normalized solutions for $\alpha$ in a suitable range.