Concentrated solutions to fractional Schrödinger-Poisson system with non-homogeneous potentials

1. Problem Statement

The paper investigates the normalized solutions (solutions with a prescribed $L^2$-mass) for the following fractional Schrödinger-Poisson system in $\mathbb{R}^3$:
$$\begin{cases} (-\Delta)^s u + V(x)u = \mu u + \phi u + a|u|^{p-2}u, \\ (-\Delta)^s \phi = u^2, \end{cases}$$
where:

• $s \in (3/4, 1)$ is the fractional order.
• $V(x)$ is a non-homogeneous, bounded external potential ($V \in L^\infty \cap C^\alpha$).
• $a > 0$ is a parameter representing the strength of the nonlinearity.
• The exponent $p$ satisfies $2 < p < 2 + \frac{4s}{3}$ (the $L^2$-subcritical to $L^2$-supercritical range, specifically focusing on the regime where the mass constraint is critical for existence).
• $\mu \in \mathbb{R}$ is the Lagrange multiplier associated with the mass constraint $\|u\|_2^2 = m$.

The system is reduced to a single nonlocal equation via the Lax-Milgram theorem:
$$(-\Delta)^s u + V(x)u = \mu u + \phi_u u + a|u|^{p-2}u,$$
where $\phi_u(x) = c_s \int_{\mathbb{R}^3} \frac{|u(y)|^2}{|x-y|^{3-2s}} dy$.

The primary objectives are to establish:

1. Existence of minimizers for the energy functional under the mass constraint.
2. Concentration behavior of these minimizers as the parameter $a \to \infty$.
3. Local uniqueness of the minimizers in the limit $a \to \infty$.

2. Methodology

The authors employ a combination of constrained variational methods, concentration-compactness principles, and asymptotic analysis tailored for nonlocal operators.

• Variational Framework: The problem is formulated as minimizing the energy functional $E_a(u)$ on the manifold $S_m = \{u \in H^s(\mathbb{R}^3) : \|u\|_2^2 = m\}$.
• Handling Non-Compactness: Since the potential $V(x)$ is bounded (not coercive like $|x|^2$), the embedding $H^s \hookrightarrow L^p$ is not compact. The authors utilize the Concentration-Compactness Lemma (Lions) to rule out vanishing and dichotomy scenarios, ensuring the existence of a minimizer.
• Energy Estimates: Precise upper and lower bounds for the minimal energy $e_m(a)$ are derived as $a \to \infty$. This involves constructing test functions based on the unique positive solution $Q$ of the limiting autonomous equation $(-\Delta)^s Q + Q - Q^{p-1} = 0$.
• Blow-up Analysis: To study concentration, the authors rescale the minimizers $u_k$ using a scaling factor $\varepsilon_k \sim a^{-\frac{2p-4}{3p-6-4s}}$. They analyze the limit profile of the rescaled functions, showing convergence to $Q$.
• Local Uniqueness via Contradiction: To prove uniqueness, the authors assume two distinct minimizers exist and define a difference function $\hat{\eta}_k$. They derive a linearized equation for $\hat{\eta}_k$ and use Pohozaev-type identities (adapted for fractional operators via the Caffarelli-Silvestre extension method) to show that the difference must vanish.
• New Techniques for Nonlocality:
  • Decay Estimates: Due to the lack of classical local elliptic estimates, the authors develop new comparison principles and use $L^\infty$ estimates combined with nonlocal term bounds to derive polynomial decay rates for the solutions and their gradients.
  • Pohozaev Identity: They overcome the difficulty of integration by parts for fractional Laplacians by using the extension method to transform the problem into a local one in higher dimensions ($\mathbb{R}^4_+$), allowing for precise integral estimates.

3. Key Contributions

1. Relaxation of Potential Conditions: Unlike previous works (e.g., [34]) that required unbounded, homogeneous potentials (like $|x|^2$), this paper establishes results for bounded, non-homogeneous potentials. The existence and concentration results depend only on the local behavior of $V(x)$ near its global minimum points.
2. Doubly Nonlocal Setting: The results address the simultaneous nonlocality of the fractional Laplacian $(-\Delta)^s$ and the Poisson interaction term $\phi_u$, a setting that introduces significant analytical difficulties compared to standard fractional Schrödinger equations.
3. Local Uniqueness in Non-Autonomous Case: The paper provides the first proof of local uniqueness for normalized solutions of the fractional Schrödinger-Poisson system in the non-autonomous (non-homogeneous potential) case.
4. Novel Analytical Tools:
   • Development of a new comparison principle to handle the decay of the difference function $\hat{\eta}_k$ in the presence of complex nonlocal terms.
   • Derivation of Pohozaev-type identities specifically for the fractional Schrödinger-Poisson system using the extension method, which is crucial for the uniqueness proof.

4. Main Results

Theorem 1.1 (Existence)

Under the assumption that $V(x) \in L^\infty \cap C^\alpha$ and $V_\infty = \lim_{|x|\to\infty} V(x) = \sup V(x)$, the minimization problem $e_m(a)$ admits at least one minimizer for any mass $m > 0$. The proof relies on the strict inequality $e_m(a) < e_\infty(a)$ to prevent vanishing at infinity.

Theorem 1.3 (Concentration Behavior)

As $a \to \infty$, the nonnegative minimizers $u_a$ concentrate around the global minimum points of $V(x)$.

• Scaling: Let $\varepsilon_a = (a/\sqrt{a^*})^{\frac{2p-4}{3p-6-4s}}$. The rescaled function $\varepsilon_a^{3/2} u_a(\varepsilon_a x + x_a)$ converges strongly in $H^s \cap L^\infty$ to $\frac{Q(x)}{\sqrt{a^*}}$, where $Q$ is the unique ground state of the limiting equation.
• Concentration Point: The maximum point $x_a$ of $u_a$ converges to a point $x_0$ where $V(x_0) = 0$.
• Refined Location: The specific location $x_0$ is determined by the local geometry of $V$ near its minima. If $V$ behaves like a homogeneous function of degree $r_i$ near a minimum $x_i$, the concentration point is selected based on minimizing a functional involving the integral of $V$ against the profile $Q^2$.

Theorem 1.4 (Local Uniqueness)

Assuming the potential $V(x)$ satisfies non-degeneracy conditions at its minimum (specifically, the set of global minima is a singleton and the Hessian of the local approximation is non-degenerate), there exists a unique nonnegative minimizer for $e_1(a)$ for sufficiently large $a$.

• The proof proceeds by contradiction, assuming two minimizers, and showing their difference $\hat{\eta}_k$ must be identically zero using the derived Pohozaev identities and the non-degeneracy of the linearized operator.

5. Significance

• Theoretical Advancement: This work bridges a gap in the literature regarding normalized solutions for fractional systems with bounded potentials. Most existing results rely on coercive potentials to ensure compactness; this paper demonstrates how to handle the lack of compactness through refined energy estimates and concentration analysis.
• Physical Relevance: The fractional Schrödinger-Poisson system models quantum many-body systems with long-range interactions (Lévy flights) and self-consistent electrostatic potentials. Understanding the behavior of these systems under mass constraints (normalized solutions) is crucial for applications in semiconductor theory and plasma physics.
• Methodological Impact: The techniques developed, particularly the adaptation of Pohozaev identities to nonlocal systems via the extension method and the new comparison principles for decay estimates, provide a robust toolkit for future research on nonlocal elliptic problems with non-homogeneous coefficients.

In summary, Liu and Yang successfully generalize previous results on normalized solutions by removing the restrictive assumptions of potential homogeneity and unboundedness, while establishing rigorous existence, concentration, and uniqueness results for a doubly nonlocal system.