Multiplicity of normalized solutions to the upper critical fractional Choquard equation with $L^2$-supercritical perturbation

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation $$(-\mathrm{\Delta }{)}^{s}u+V(\epsilon x)u=\lambda u+(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+(I_{\alpha }\ast \mathrm{\mid }u{\mathrm{\mid }}^q)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u\text{in }{\mathbb{R}}^N,$$ where $N>2s$, $0<s<1$, $(N-4s{)}^{+}<\alpha <N$, and the nonlocal exponents satisfy $$\frac{N+2s+\alpha }{N}<q<p=\frac{N+\alpha }{N-2s},$$ so that both nonlinearities are $L^2$-supercritical and the $p$ term has upper critical growth of Hartree type. Under standard assumptions on the slowly varying potential $V$, we develop a constrained variational approach on the $L^2$-sphere, based on a truncation-penalization of the critical term in the energy functional, to overcome the lack of compactness. We prove that, for all sufficiently small $\epsilon >0$, the problem admits at least ${\mathrm{c}\mathrm{a}\mathrm{t}}_{M_{\delta }}(M)$ distinct normalized solutions, where $M$ is the set of global minima of $V$ and these solutions concentrate near $M$ as $\epsilon \to 0$.