Critical Schrödinger-Bopp-Podolsky System

• The Critical Schrödinger-Bopp-Podolsky system is a set of nonlinear elliptic and dispersive PDEs modeling standing wave quantum states coupled with a Bopp–Podolsky electrostatic field at the Sobolev critical growth threshold.
• It employs advanced variational methods, including Mountain–Pass techniques and concentration–compactness, to address noncompactness and critical exponent challenges.
• Analytical results focus on existence, multiplicity, and ground state characterization, with implications for quantum electrostatics and nonlinear field models.

The Critical Schrödinger–Bopp–Podolsky (SBP) system refers to a class of nonlinear elliptic and dispersive partial differential equations modeling standing wave states of a quantum particle coupled to a Bopp–Podolsky type electrostatic field, in the presence of nonlinearities at the Sobolev critical growth threshold. In spatial dimension three, the system generically takes the form

$$\begin{cases} -\Delta u + q^2 \phi u = |u|^{p-2}u + |u|^4 u, \ -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2, \end{cases}$$

for unknowns $u,\phi:\mathbb{R}^3\to\mathbb{R}$ (or on bounded domains) with parameters $a>0$ (Bopp-Podolsky length scale), $q\neq0$ (coupling constant), and a nonlinear exponent $p$ straddling the Sobolev critical value $p=6$. Prominent variants include mass constraints, sublinear perturbations, Choquard-type nonlocal terms, and domain/boundary settings. The critical nonlinearities induce substantial analytical complexities, necessitating refined variational and compactness techniques.

1. Formulations and Structural Properties

The canonical SBP system arises from coupling the nonlinear stationary Schrödinger equation with the (generalized) Bopp–Podolsky electrostatic field. The essential equations on $\mathbb{R}^3$ are: $$-\Delta u + \omega u + q^2 \phi u = |u|^{p-2}u, \qquad -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2$$ for $\omega>0, a>0$ (d'Avenia et al., 2018). The nonlocal field $\phi$ satisfies a fourth order elliptic equation, with $a$ encoding a physical cutoff/fundamental length.

Criticality emerges at the Sobolev exponent $p=6$ where embeddings transition from compact to merely continuous. When mass constraints are imposed ($\int u^2 dx = c$), or additional critical/local/nonlocal nonlinearities enter, the system's geometry and compactness properties are dramatically altered (Li et al., 2023, Huang et al., 2024, Huang et al., 27 Oct 2025, Chen et al., 3 Jan 2026, Damian et al., 25 Jul 2025).

Key function spaces:

• $u$ in $H^1(\mathbb{R}^3)$ or $D^{1,2}$ (zero-mass case)
• $\phi$ in $$D = \{\varphi \in D^{1,2} : \Delta\varphi \in L^2\}$$

The electrostatic potential can be explicitly represented by convolution with the Bopp–Podolsky kernel $K(x) = (1 - e^{ -|x|/a }) / |x|$.

2. Critical Exponents and Variational Frameworks

The system's analytical setup is dictated by the interplay of nonlinear growth and the underlying functional framework. The Sobolev critical exponent in $\mathbb{R}^3$ is $p_c = 6$; this is pivotal:

• For $2 < p < 6$ (subcritical), standard variational methods (Mountain Pass, minimization) yield existence theorems for small charge and/or mass (d'Avenia et al., 2018, Li et al., 2023).
• At $p=6$ or $p > 6$ (critical/supercritical), Pohožaev-type nonexistence prevails in the unconstrained case; mass constraints or zero-mass settings necessitate novel functional constructions and compactness results (Huang et al., 27 Oct 2025).

Reduced energy functionals take the form

$$J(u) = \frac{1}{2} \|u\|_{H^1}^2 + \frac{q^2}{4} \int \phi_u u^2 - \frac{\mu}{p} \int |u|^p - \frac{1}{6} \int |u|^6,$$

where $\phi_u$ is determined by convolution.

For systems incorporating mass constraints, the energy is constrained to manifolds such as $S(c) = \{u \in H^1: \|u\|_2^2 = c\}$ and minimization/critical point searches are performed under Lagrange multipliers.

Noncompactness at criticality ($p=6$) is addressed via:

• Concentration–compactness
• Truncation and fiber map analysis (Li et al., 2023, Huang et al., 27 Oct 2025, Chen et al., 3 Jan 2026)
• Genus theory and minimax schemes for multiplicity (Chen et al., 3 Jan 2026, Damian et al., 25 Jul 2025)

3. Existence, Multiplicity, and Ground State Results

Existence results are sensitive to the regime:

• In $\mathbb{R}^3$, with $2 < p < 6$, nontrivial solutions exist for small charge or arbitrary $p>3$ via Mountain–Pass (d'Avenia et al., 2018).
• For zero-mass SBP systems, existence of positive ground states is proved for $p \in (3,6)$ with critical term $|u|^4u$; multiplicity is shown for $p \in (4,6)$ through abstract index theory (Huang et al., 27 Oct 2025).
• On bounded domains with critical Choquard-type nonlinearity, infinitely many normalized solutions emerge via genus and minimax constructions, subject to small mass thresholds (Chen et al., 3 Jan 2026).
• With sublinear (power $p\in(0,1)$) perturbations and critical terms, an infinite sequence of negative energy solutions appears, including ground states, for sufficiently small sublinear coefficient (Damian et al., 25 Jul 2025).

Compact embedding in radial spaces, strong subadditivity for mass splitting, and Pohožaev identities are central for establishing ground state minimizers and concentration phenomena (Li et al., 2023, Huang et al., 27 Oct 2025).

4. Palais–Smale Compactness, Concentration, and Limiting Behavior

Criticality induces potential loss of compactness; precompactness and convergence of (Palais–Smale) sequences require fine estimates:

• Palais–Smale holds below specific energy levels tied to sharp constants (e.g., $c < (1/3) S^{3/2}$ where $S$ is the Sobolev constant) (Huang et al., 27 Oct 2025, Damian et al., 25 Jul 2025).
• Constrained minimization and truncation techniques are used to localize energy (Chen et al., 3 Jan 2026).
• Concentration–compactness analyzes splitting/dichotomy, particularly relevant for mass-constrained and bounded domain problems (Li et al., 2023, Huang et al., 27 Oct 2025).

Limiting regimes are well-characterized:

• As $a \rightarrow 0$, the SBP system solutions converge to those of the classical Schrödinger–Poisson system in $H^1$ and $D^{1,2}$; the Bopp–Podolsky kernel transitions to the Coulomb potential (d'Avenia et al., 2018, Damian et al., 25 Jul 2025).
• As mass, sublinear coefficient, or critical parameters vanish, solutions concentrate and may collapse to zero or classical NLS ground states, with explicit scaling laws and profile description (Huang et al., 2024, Damian et al., 25 Jul 2025).
• The mountain-pass energy level yields a threshold for global existence versus finite time blow-up in the corresponding time-dependent SBP equation (Huang et al., 2024).

5. Nonlinearities: Sublinear, Critical, and Nonlocal Terms

Beyond the pure power case, recent works address systems with:

• Sublinear local perturbations $\lambda K(x)|u|^{p-1}u$, admitting infinite negative energy solutions and collapse phenomena for vanishing $\lambda$ (Damian et al., 25 Jul 2025).
• Critical nonlocal Choquard-type terms $(I_\alpha*|u|^{3+\alpha})|u|^{1+\alpha}u$, analyzed on bounded domains via genus and minimax approaches under mass constraints (Chen et al., 3 Jan 2026).
• Combined critical and subcritical nonlinearities yielding richer variational geometry and solution multiplicity (Li et al., 2023, Huang et al., 27 Oct 2025).

These generalizations provoke further compactness issues and necessitate functional truncations, abstract index theory, and careful scrutiny of regularity/decay properties.

6. Analytical Tools and Key Estimates

The study of critical SBP systems leverages several advanced analytical methods:

• Mountain–Pass Theorem, Ekeland’s variational principle
• Truncation via Jeanjean–Le Coz methods for subcritical cases
• Concentration–compactness and splitting lemmas in critical and zero-mass regimes
• Coercive and compact embeddings in radial/zero-mass spaces (e.g., $D^{1,2} \hookrightarrow L^6$)
• Pohožaev and Nehari-type identities for existence/nonexistence and energy minimization (d'Avenia et al., 2018, Huang et al., 27 Oct 2025)
• Abstract genus theory for multiplicity results in symmetric settings (Damian et al., 25 Jul 2025, Chen et al., 3 Jan 2026)
• Uniform estimates on the Bopp–Podolsky Green’s function for convolution representations

$$0 \leq K(x) \leq |x|^{-1}, \quad |\nabla K(x)| \lesssim |x|^{-2}, \quad |\Delta K(x)| \lesssim |x|^{-3}$$

7. Open Problems and Qualitative Features

Current challenges include:

• Extension to $L^2$-critical or supercritical regimes and the effect on solution multiplicity, orbital stability, and blow-up
• Precise energy thresholds and mass concentration dynamics under scaling limits (Huang et al., 2024)
• Uniqueness, symmetry, and decay characterization of ground states in general critical SBP systems (Li et al., 2023, Huang et al., 2024)
• Systems on bounded domains with general boundary conditions, sign-changing potentials, and noncompact embeddings (Chen et al., 3 Jan 2026)
• Stability analysis and spectral properties beyond the variational setting

A plausible implication is that advancements in compactness theory and abstract index frameworks will catalyze deeper understanding of critical growth phenomena, bifurcation structures, and dynamical stability in SBP and hybrid local–nonlocal models.

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Key References:

• D'Avenia, Siciliano, "Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case" (d'Avenia et al., 2018)
• Huang, Wang, "Existence and multiplicity results for the zero mass Schrödinger-Bopp-Podolsky system with critical growth" (Huang et al., 27 Oct 2025)
• Huang, Wang, "Normalized ground states for the mass supercritical Schrödinger-Bopp-Podolsky system" (Huang et al., 2024)
• Wang et al., "Normalized Solutions for Schrödinger-Bopp-Podolsky Systems with Critical Choquard-Type Nonlinearity on Bounded Domains" (Chen et al., 3 Jan 2026)
• Wang, "Schrödinger-Bopp-Podolsky system with sublinear and critical nonlinearities: solutions at negative energy levels and asymptotic behaviour" (Damian et al., 25 Jul 2025)
• Wen, Huang, Sun, "Normalized solutions for Sobolev critical Schrödinger-Bopp-Podolsky systems" (Li et al., 2023)