Normalized Semiclassical Solutions to Magnetic Schrödinger-Poisson Systems with Critical Local and Nonlocal Interactions

We study the existence, multiplicity, and concentration of normalized semiclassical states for a magnetic Schrödinger–Poisson system in ${\mathbb{R}}^3$ featuring both the Sobolev-critical local nonlinearity $\mathrm{\mid }u{\mathrm{\mid }}^{4}u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=a^2{\epsilon }^3,$ where $a>0$ denotes the prescribed mass and $\epsilon >0$ is the semiclassical parameter. By combining constrained variational methods, a suitable penalization scheme, concentration–compactness arguments, and Ljusternik–Schnirelmann theory, we first prove the existence of a normalized semiclassical solution for sufficiently small $a$ and $\epsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\epsilon >0$, the number of distinct normalized solutions is bounded from below by the Ljusternik–Schnirelmann category of the minimum set $$\mathcal{M}=\{x\in {\mathbb{R}}^3:V(x)=\underset{{\mathbb{R}}^3}{\min }V\}\mathrm{.}$$ Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal{M}$ as $\epsilon \to 0$.