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

Abstract

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 \(\varepsilon\to0\).