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).

• The paper establishes the existence and multiplicity of normalized ground states using refined variational methods and topological arguments.
• It analyzes semiclassical concentration by linking solution localization to the minimum set of the electric potential.
• The work derives sharp compactness thresholds by effectively handling Hardy singularities along with critical local and nonlocal nonlinearities.

Summary of "Normalized states for a critical Schrödinger--Poisson system with Hardy singularity: multiplicity and semiclassical concentration" (2607.11277)

Problem Setting and Motivation

The paper investigates the existence, multiplicity, and concentration of normalized solutions for a semiclassical critical nonlinear Schrödinger–Poisson system in $\mathbb{R}^3$ with an inverse-square Hardy singularity. The studied system consists of a Schrödinger equation featuring a singular attractive Hardy potential, critical local and nonlocal nonlinearities, and a mass constraint, coupled with a Poisson equation modeling the electrostatic potential induced by the wave function density.

Explicitly, the main problem is: $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$ where $\varepsilon > 0$ is a semiclassical parameter, $0<\kappa<1/4$ ensures Hardy coercivity, $q \in (2,10/3)$ guarantees subcritical growth, and $V$ is a continuous electric potential with a compact minimum set $\mathcal M$. The mass constraint is physically motivated by fixing the $L^2$-norm representing the particle number.

The system's analytical challenge arises from the simultaneous treatment of mass constraint, attractive Hardy singularity, Sobolev-critical local nonlinearity, and critical nonlocal Poisson interaction, which fundamentally alters the variational framework compared to the classical Schrödinger–Poisson setting.

Variational Framework and Analytical Tools

A reduced energy functional is constructed on the constraint sphere $S_{a,\varepsilon}$: $$J_\varepsilon(u) = \frac{1}{2}\|u\|_{\varepsilon}^2 - \frac{1}{10} \int_{\mathbb R^3} \phi_u |u|^5 dx - \frac{\mu}{q} \|u\|_q^q - \frac{1}{6} \|u\|_6^6,$$ where the norm $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$0 incorporates the Hardy potential: $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$1

The Poisson interaction is reduced via the Newton potential and Hardy–Littlewood–Sobolev estimates. The scaling properties under the mass-preserving transformation are central, particularly in the analysis of concentration phenomena in the semiclassical limit $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$2. The Pohozaev identity and its associated manifold play a vital role in characterizing normalized solutions.

To handle lack of compactness due to critical nonlinearities and the singularity, the paper leverages Hardy’s inequality, Gagliardo–Nirenberg estimates, truncation methods, and concentration–compactness principles. The variational setting ensures the mountain-pass geometry for the constrained functional under small mass and small $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$3.

Main Results

Existence of Ground States

The first main theorem establishes existence of a normalized ground state $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$4 (positive away from the singularity) for sufficiently small mass $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$5 and semiclassical parameter $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$6, achieved via mountain-pass minimax and refined compactness analysis. The proof relies crucially on the coercivity imposed by the Hardy inequality: $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$7

Multiplicity via Lusternik–Schnirelmann Category

The second main theorem establishes that the number of positive normalized solutions is bounded below by the Lusternik–Schnirelmann category of the potential’s minimum set $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$8. A barycenter map, constructed on low-energy constrained states, is shown to transfer the topological complexity of $$\begin{cases} -\varepsilon^2\Delta u+\left( V(x)-\frac{\kappa\varepsilon^2}{|x|^2} \right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u,\ -\varepsilon^2\Delta\phi=|u|^5,\ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \end{cases}$$9 to the variational landscape, yielding multiplicity of solutions.

Semiclassical Concentration

It is demonstrated that, as $\varepsilon > 0$0, semiclassical states concentrate near the global minimum set $\varepsilon > 0$1 of the electric potential. After appropriate rescaling, the concentrating sequences strongly converge in $\varepsilon > 0$2 to ground states of an autonomous limit problem without the Hardy singularity, provided the concentration region is separated from the origin.

Sharp Compactness Thresholds

Distinction is made between regular and singular bubbling scenarios. Two critical thresholds are derived: $\varepsilon > 0$3 for regular (away from the singularity) bubbling, and $\varepsilon > 0$4 for singular (near-origin) bubbling, leading to a mixed compactness threshold $\varepsilon > 0$5. Palais–Smale sequences below this threshold are shown to be compact.

Numerical and Structural Highlights

• Strong claim: The paper provides a lower bound for the number of normalized positive solutions by the category of the minimum set, and confirms that such solutions concentrate near $\varepsilon > 0$6 as $\varepsilon > 0$7.
• Structural novelty: Combining a prescribed mass constraint, inverse-square Hardy singularity, critical local and nonlocal nonlinearities, and semiclassical scaling yields a variational geometry substantially richer and more delicate than previously studied non-magnetic or non-singular Schrödinger–Poisson models.
• Quantitative estimates: Uniform bounds for energy and solution norms are established for Palais–Smale–Pohozaev sequences, independent of $\varepsilon > 0$8 for small values.

Implications and Future Directions

From a theoretical perspective, the results deepen understanding of nonlinear Schrödinger–Poisson systems with singular potentials and critical nonlinearities, revealing new mechanisms for existence, multiplicity, and concentration under normalization constraints. This framework likely applies to models in quantum mechanics and plasma physics where such singular interactions and normalization are physically significant.

The techniques, especially the variational handling of both mass constraints and Hardy singularities, suggest further study of normalized states for more general nonlocal elliptic systems. Extensions to fractional Laplacians, magnetic fields, and other singular coefficients are expected to be fruitful, as valued in recent related works. Topological multiplicity results may encourage further investigation of phenomena where the topology of the concentration region governs solution structure.

On the practical side, the concentration phenomena may inform numerical approaches for semiclassical states, as the explicit localization near minima provides guidance for computational schemes.

Conclusion

The paper rigorously establishes existence, multiplicity, and semiclassical concentration of normalized critical states for a singular Schrödinger–Poisson system with Hardy potential. By combining advanced variational, topological, and compactness tools, it clarifies how mass constraint, singularity, and critical nonlinearity interact, linking the topology of the potential’s minimum set with the multiplicity of concentrating normalized solutions. This work provides a robust foundation for future analysis and applications of nonlinear systems with singular interactions and constraints.