Normalized solutions of quasilinear Schrödinger-Poisson system with critical nonlinear term in bounded domain

1. Problem Statement

The paper investigates the existence and multiplicity of normalized solutions for a coupled quasilinear Schrödinger-Poisson system defined on a bounded domain $\Omega_r \subset \mathbb{R}^3$. The system is given by:

$$\begin{cases} -\Delta u + \phi u + \lambda u = |u|^{q-2}u + |u|^4u, & x \in \Omega_r, \\ -\Delta \phi - \varepsilon^4 \Delta_4 \phi = u^2, & x \in \Omega_r, \\ u = \phi = 0, & x \in \partial \Omega_r, \end{cases}$$

subject to the mass constraint (normalization condition):
$$\int_{\Omega_r} |u|^2 dx = b^2,$$
where:

• $\lambda$ is an unknown frequency parameter (Lagrange multiplier) emerging from the constraint.
• $\varepsilon > 0$ is a parameter associated with the quasilinear term.
• $\Delta_4 \phi = \text{div}(|\nabla \phi|^2 \nabla \phi)$ is the 4-Laplacian operator.
• The nonlinearity includes a critical exponent term ($|u|^4u$, corresponding to the Sobolev critical exponent $2^*=6$ in $\mathbb{R}^3$) and a subcritical term $|u|^{q-2}u$ with $q \in (2, 8/3)$.

Physical Context: The system models quantum phenomena in semiconductor nanostructures, specifically interactions between quantum confinement and longitudinal electric field oscillations. The term $\varepsilon^4 \Delta_4 \phi$ represents a nonlinear correction to the standard Poisson equation.

2. Methodology

The authors employ a combination of variational methods, topological tools, and compactness analysis to overcome two primary difficulties: the lack of a standard extremal geometric structure for the energy functional and the loss of compactness due to the critical exponent.

A. Reduction to a Single-Variable Functional

Following the reduction method, the authors first solve the quasilinear Poisson equation for $\phi$ in terms of $u$. For a fixed $u$, there exists a unique solution $\phi_\varepsilon(u) \in H^1_0(\Omega_r) \cap W^{1,4}_0(\Omega_r)$. This allows the system to be reduced to a single functional $E(u)$ defined on the constraint manifold $S_{r,b} = \{u \in H^1_0(\Omega_r) : \|u\|_2 = b\}$.

B. Truncation Technique

Due to the critical exponent ($|u|^6$), the energy functional does not satisfy the Palais-Smale (PS) condition globally. To address this:

1. Geometric Analysis: The authors analyze the behavior of the functional $G(t)$ (related to the energy) and show that for sufficiently small mass $b$, the functional admits a local positive maximum.
2. Cutoff Function: A smooth cutoff function $T(\|\nabla u\|_2)$ is introduced to truncate the critical term $|u|^6$ when the gradient norm exceeds a certain threshold. This creates a modified functional $E_T(u)$ that coincides with the original functional in a specific energy range but restores the necessary compactness properties for the PS condition.

C. Genus Theory and Multiplicity

To prove the existence of multiple solutions (rather than just one), the authors utilize Krasnoselskii's genus theory:

• They define a family of symmetric subsets on the constraint manifold $S_{r,b}$.
• Using the minimax principle, they construct a sequence of critical values $d_k$.
• Proposition 2.1 (based on genus theory) guarantees that if the genus of the level sets is sufficiently high, there exist at least $k$ distinct pairs of solutions.

D. Concentration-Compactness Principle

To handle the critical exponent and prove that the truncated solutions are indeed solutions to the original problem, the authors invoke Lions' Concentration-Compactness Principle.

• They analyze the behavior of Palais-Smale sequences.
• They prove that the index set of concentration points is empty by deriving a contradiction involving the Sobolev constant $S$ and the bounds on the gradient norm.
• This ensures that the weak limit of the sequence is a strong limit, recovering the critical term $|u|^6$ without loss of mass.

E. Asymptotic Analysis

For Theorem 1.2, the authors study the limit as $\varepsilon \to 0$. They demonstrate that the solutions of the quasilinear system converge strongly to the solutions of the classical Schrödinger-Poisson system (where the Poisson equation becomes linear: $-\Delta \phi = u^2$).

3. Key Results

Theorem 1.1: Multiplicity of Normalized Solutions

For any fixed integer $k \in \mathbb{N}$, there exists a threshold mass $b^* > 0$ such that for any $b \in (0, b^*)$, the system possesses at least $k$ distinct normalized solutions $(u_j, \phi_j, \lambda_j)$.

• Significance: This establishes the existence of infinitely many solutions (by varying $k$) for small mass constraints, a result that is non-trivial due to the critical growth and the quasilinear nature of the system.

Theorem 1.2: Asymptotic Convergence

As the parameter $\varepsilon \to 0$, the family of solutions $(u_{j,\varepsilon}, \phi_{j,\varepsilon}, \lambda_{j,\varepsilon})$ converges strongly (in $H^1_0$ and $L^6$) to a solution $(u_{j,0}, \phi_{j,0}, \lambda_{j,0})$ of the classical Schrödinger-Poisson system:
$$\begin{cases} -\Delta u + \phi u = \lambda u + |u|^{q-2}u + |u|^4u, \\ -\Delta \phi = u^2, \\ \int |u|^2 = b^2. \end{cases}$$

• Significance: This validates the physical consistency of the model, showing that the quasilinear correction vanishes smoothly to recover the standard model.

4. Significance and Contributions

1. Handling Critical Growth in Bounded Domains: The paper successfully addresses the lack of compactness caused by the critical exponent $|u|^4u$ in a bounded domain setting, a scenario often more complex than the whole space $\mathbb{R}^3$ due to boundary effects.
2. Normalized Solutions: Unlike many previous works that fix the frequency $\lambda$ and find solutions, this work treats $\lambda$ as an unknown determined by the mass constraint $b$. This is physically more relevant for modeling systems with fixed particle numbers.
3. Overcoming Non-Extremal Geometry: The authors circumvent the difficulty that the energy functional lacks a standard mountain-pass geometry for normalized problems by restricting the exponent $q$ to $(2, 8/3)$ and using a truncation method combined with genus theory.
4. Extension of Literature: The results extend previous studies on quasilinear Schrödinger-Poisson systems (which often focused on existence of ground states or unbounded domains) to the multiplicity of normalized solutions in bounded domains with critical nonlinearities.

5. Conclusion

The paper provides a rigorous mathematical framework for analyzing quasilinear Schrödinger-Poisson systems with critical nonlinearities under mass constraints. By synthesizing truncation techniques, genus theory, and concentration-compactness principles, the authors prove the existence of multiple normalized solutions and establish their convergence to the classical linear case, significantly advancing the understanding of these coupled quantum systems.