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 Schrödinger–Poisson system with a singular Hardy potential in ${\mathbb{R}}^3$. More precisely, we consider $$\{\begin{array}{ll}-{\epsilon }^2\mathrm{\Delta }u+(V(x)-{\displaystyle \frac{\kappa {\epsilon }^2}{\mathrm{\mid }x{\mathrm{\mid }}^2}})u-\phi \mathrm{\mid }u{\mathrm{\mid }}^{3}u=\lambda u+\mu \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{4}u,& \text{in }{\mathbb{R}}^3,\\ -{\epsilon }^2\mathrm{\Delta }\phi =\mathrm{\mid }u{\mathrm{\mid }}^5,& \text{in }{\mathbb{R}}^3,\end{array}$$ under the prescribed mass constraint $${\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=a^2{\epsilon }^3,$$ where $a,\mu >0$, $q\in (2,10\mathrm{/}3)$, $\epsilon >0$ is a small semiclassical parameter, and $0<\kappa <1\mathrm{/}4$. The parameter $\lambda \in \mathbb{R}$ appears as a Lagrange multiplier associated with the mass constraint, while $V:{\mathbb{R}}^3\to (0,+\mathrm{\infty })$ 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 $\epsilon$. 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 $\epsilon \to 0$.