Khaled Khachnaoui | Searcharxiv

arXiv2026

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 \[ \begin{cases} -\varepsilon^2Δu+ \left(V(x)-\dfrac{κ\varepsilon^2}{|x|^2}\right)u -ϕ|u|^3u =λu+μ|u|^{q-2}u+|u|^4u, & \text{in } \mathbb{R}^3, \\[1mm] -\varepsilon^2Δϕ=|u|^5, & \text{in } \mathbb{R}^3, \end{cases} \] under the prescribed mass constraint \[ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \] where $a,μ>0$, $q\in(2,10/3)$, $\varepsilon>0$ is a small semiclassical parameter, and $0<κ<1/4$. The parameter $λ\in\mathbb{R}$ 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 $\varepsilon\to0$.

math.AP↗

arXiv2026

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 $|u|^4u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint $\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3,$ where $a>0$ denotes the prescribed mass and $\varepsilon>0$ 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 $\varepsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\varepsilon>0$, 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 $\varepsilon\to0$.

math.AP↗