Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation

In this paper, we study the mass-constrained fractional Choquard equation $(-\mathrm{\Delta }{)}^{s}u=\lambda u+\alpha (I_{\mu }\ast \mathrm{\mid }u{\mathrm{\mid }}^{\frac{2N-\mu }{N}})\mathrm{\mid }u{\mathrm{\mid }}^{\frac{2N-\mu }{N}-2}u+(I_{\mu }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in ${\mathbb{R}}^N$, under the constraint ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=c^2>0$, where $N>2s$, $s\in (0,1)$, $\mu \in (0,N)$, $\alpha >0$, and $2+\frac{2s-\mu }{N}\le p<\frac{2N-\mu }{N-2s}$. We first establish a nonexistence result in the $L^2$-critical case $p=2+\frac{2s-\mu }{N}$. Then, in the $L^2$-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity ${\mathcal{M}}_1(c)$. Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps.