Normalized states for a critical Schrödinger--Poisson system with Hardy singularity: multiplicity and semiclassical concentration

In this paper, we investigate the existence, multiplicity, and semiclassical concentration of normalized solutions to a critical Schr\"{o}dinger--Poisson system with a singular Hardy potential in $\mathbb{R}^3$. More precisely, we consider $$\begin{cases} -\varepsilon^2\Delta u+ \left(V(x)-\dfrac{\kappa\varepsilon^2}{|x|^2}\right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u, & \text{in } \mathbb{R}^3, \\[1mm] -\varepsilon^2\Delta\phi=|u|^5, & \text{in } \mathbb{R}^3, \end{cases}$$ under the prescribed mass constraint $$\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3,$$ where $a,\mu>0$, $q\in(2,10/3)$, $\varepsilon>0$ is a small semiclassical parameter, and $0<\kappa<1/4$. The parameter $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier associated with the mass constraint, while $V:\mathbb{R}^3\to(0,+\infty)$ is a continuous electric potential whose minimum set is assumed to be nonempty and compact. The main difficulty stems from the simultaneous presence of the inverse-square Hardy singularity, the mass constraint, the critical local nonlinearity, and the nonlocal Poisson interaction. By combining the Hardy inequality, constrained variational methods, suitable truncation arguments, and concentration-compactness techniques, we first establish the existence of a normalized ground state for sufficiently small mass and sufficiently small $\varepsilon$. We then employ Ljusternik--Schnirelmann category theory to obtain multiple normalized solutions whose number is related to the topology of the minimum set of $V$. Finally, we show that the corresponding semiclassical states concentrate near the global minimum set of the electric potential as $\varepsilon\to0$.