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

Abstract

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 $|u{|}^{4}u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint ${\int }_{{\mathbb{R}}^3}|u{|}^{2}dx=a^2{\epsilon }^3,$ 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 \[ \mathcal M = \{x\in\mathbb{R}^3:V(x)=\min_{\mathbb{R}^3}V\}. \] Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal{M}$ as $\epsilon \to 0$.