The uniqueness and concentration behavior of solutions for a nonlinear fractional Schrödinger system

Technical Summary: Uniqueness and Concentration Behavior of Solutions for a Nonlinear Fractional Schrödinger System

Problem Statement
This paper investigates the uniqueness and concentration behavior of solutions for a nonlinear system of two coupled fractional Schrödinger equations in $\mathbb{R}$:
$$\begin{cases} \sqrt{-\Delta}u_1 + V_1(x)u_1 = \mu_1 u_1 + a_1 u_1^3 + \beta u_2^2 u_1, \\ \sqrt{-\Delta}u_2 + V_2(x)u_2 = \mu_2 u_2 + a_2 u_2^3 + \beta u_1^2 u_2, \end{cases}$$
where $\sqrt{-\Delta}$ denotes the fractional Laplacian of order $1/2$ (defined via Fourier transform), $V_i(x)$ are trapping potentials, and the parameters $a_1, a_2, \beta > 0$ represent attractive intraspecies and interspecies interactions, respectively. The study focuses on $L^2$-normalized solutions (mass constraint $\int u_i^2 = 1$) where the chemical potentials $\mu_i$ act as Lagrange multipliers.

The system models non-Gaussian Bose-Einstein condensates (BEC) where bosons follow Lévy flights. While existence and non-existence results for this system were previously established in [17], the uniqueness of minimizers and the detailed asymptotic behavior of solutions as interaction strengths approach a critical threshold remained unexplored.

Methodology
The authors employ variational methods combined with asymptotic analysis and the implicit function theorem. The core approach involves:

1. Variational Framework: The problem is formulated as a constrained minimization problem for an energy functional $E_{a_1, a_2, \beta}$ subject to mass constraints. The existence of minimizers is linked to the fractional Gagliardo-Nirenberg inequality.
2. Uniqueness via Implicit Function Theorem: To prove uniqueness for small interaction parameters, the authors analyze the linearized operator around the ground state of the associated linear problem. They utilize the spectral properties of the operator $\sqrt{-\Delta} + V_i(x)$ and apply the implicit function theorem in a suitable Banach space setting to show that the minimizer is unique and non-negative when parameters are sufficiently small.
3. Concentration Analysis: To study the behavior as interaction strengths $(a_1, a_2)$ approach the critical threshold $(a^*-\beta, a^*-\beta)$ (where $a^* = \|Q\|_2^2$ and $Q$ is the ground state of the limiting equation), the authors:
   • Derive optimal energy estimates for the minimizers.
   • Utilize a rescaling technique involving a parameter $\epsilon$ related to the distance from the critical threshold.
   • Analyze the limiting profile of the rescaled solutions, showing they converge to the ground state of the limiting equation.
   • Address the non-local nature of the fractional Laplacian, which prevents the direct application of the strong maximum principle used in classical Laplacian systems. Instead, they deduce positivity from the strict positivity of the resolvent kernel.
   • Handle the polynomial decay of the fractional ground state (as opposed to exponential decay in the classical case) by imposing specific conditions on the growth of the trapping potentials near their minima.

Key Contributions and Results

1. Uniqueness of Minimizers (Theorem 1.1):
   The paper proves that if the trapping potentials satisfy standard conditions (local boundedness, coercivity), the minimization problem admits a unique non-negative minimizer provided the interaction parameters $(a_1, a_2, \beta)$ are sufficiently small. This extends previous uniqueness results from single-component equations to the coupled fractional system.

2. Concentration Behavior (Theorem 1.3):
   As the total interaction strength approaches the critical value, the paper establishes that:

   • Blow-up: Each component of the solution blows up (its $L^\infty$ norm tends to infinity).
   • Concentration Point: The solutions concentrate at a "flattest" common minimum point of the trapping potentials $V_1(x) + V_2(x)$. Specifically, if the potentials behave like $|x-x_{ij}|^{p_{ij}}$ near their minima, the concentration occurs at the points where the exponent $p_{ij}$ is minimized (the "flattest" wells).
   • Optimal Blow-up Rate: The paper provides the precise rate at which the solutions blow up and the rate at which the concentration points converge to the global minimum.
   • Limit Profile: Under appropriate rescaling, the components of the solution converge strongly in $H^{1/2}(\mathbb{R})$ to a scaled version of the unique radially symmetric ground state $Q$ of the limiting equation $\sqrt{-\Delta}u + u = u^3$.

3. Symmetry Breaking:
   The authors demonstrate that even if the trapping potentials possess symmetry (e.g., $V_1 = V_2$ with multiple symmetric minima), the uniqueness of the minimizer can break down as the interaction strength increases. In the critical regime, the system admits multiple minimizers, each concentrating at a different minimum of the potential, indicating symmetry breaking.

Significance and Claims
The paper claims to fill a critical gap in the literature regarding the fractional Schrödinger system with coupled components. While existence was known, the uniqueness and concentration behavior were open problems.

• Novelty in Non-local Analysis: The authors highlight that the presence of the non-local operator $\sqrt{-\Delta}$ introduces significant difficulties compared to classical Laplacian systems. Specifically, the strong maximum principle does not apply directly to establish positivity. The authors overcome this by leveraging the strict positivity of the resolvent kernel.
• Handling Polynomial Decay: Unlike classical Schrödinger equations where ground states decay exponentially, the fractional ground state decays polynomially ($O(|x|^{-2})$). This necessitates specific constraints on the trapping potentials (specifically, the order of the potential near minima must be less than 3, and for optimal estimates, less than 1) to ensure integrability and valid energy estimates.
• Physical Relevance: The results provide a rigorous mathematical foundation for understanding the stability and spatial distribution of two-component non-Gaussian Bose-Einstein condensates, particularly how they behave near critical interaction strengths.

The paper concludes that the uniqueness of the ground state for the limiting equation is the key factor enabling the proof of uniqueness and concentration for the coupled system, despite the non-uniqueness of positive solutions to the intermediate scaled equations. The case where the fractional order $s \neq 1/2$ remains an open problem, as noted in Remark 1.5.