Segregated solutions for a critical Choquard system with a small interspecies repulsive force

1. Problem Statement

The paper investigates the existence of segregated solutions for a coupled system of critical elliptic equations with nonlocal nonlinearities of Choquard type in $\mathbb{R}^4$. The system is given by:

$$\begin{cases} -\Delta u = (|x|^{-\mu} * |u|^{\frac{4-\mu}{2}})|u|^{\frac{2-\mu}{2}}u + \beta uv^2 & \text{in } \mathbb{R}^4 \\ -\Delta v = (|x|^{-\mu} * |v|^{\frac{4-\mu}{2}})|v|^{\frac{2-\mu}{2}}v + \beta u^2v & \text{in } \mathbb{R}^4 \end{cases}$$

Key Parameters and Context:

• Dimension: $N=4$.
• Nonlocal Term: The convolution term $|x|^{-\mu} * |u|^p$ corresponds to the Hardy-Littlewood-Sobolev (HLS) inequality, where $0 < \mu \le 4$.
• Criticality: The exponents correspond to the "upper critical exponent" for the Choquard equation, $2^*_\mu = \frac{2N-\mu}{N-2}$. In $\mathbb{R}^4$, this makes the problem critical with respect to the Sobolev embedding.
• Coupling: The parameter $\beta < 0$ represents a competitive (repulsive) interaction between the two components $u$ and $v$.
• Goal: To construct solutions where the first component ($u$) resembles a single radial ground state, while the second component ($v$) exhibits a "blow-up" behavior, concentrating at $k$ distinct points arranged as vertices of a regular polygon.

2. Methodology

The author employs the Lyapunov-Schmidt reduction method, a perturbative technique used to reduce an infinite-dimensional problem to a finite-dimensional one. The process involves the following steps:

1. Ansatz Construction:

   • The solution is approximated by a superposition of known solutions to the limit problems.
   • Component $u$: Approximated by the unique positive radial solution $U$ of the single critical Choquard equation.
   • Component $v$: Approximated by a sum of $k$ bubbles $V = \sum_{j=1}^k U_{\lambda, \xi_j}$, where $U_{\lambda, \xi_j}$ are rescaled and translated versions of the ground state.
   • Geometry: The concentration points $\xi_j$ are placed at the vertices of a regular polygon in the $x_1, x_2$ plane: $\xi_j = \rho (\cos \frac{2\pi(j-1)}{k}, \sin \frac{2\pi(j-1)}{k}, 0, 0)$.
   • Scaling: The parameter $\lambda$ (related to the concentration scale) is chosen such that $\frac{1}{\lambda} \to \infty$ as $|\beta| \to 0$.

2. Functional Setting and Symmetries:

   • The problem is set in a weighted space with norms $\|\cdot\|_*$ and $\|\cdot\|_{**}$ to control the decay of error terms.
   • Symmetry Constraints: To handle the kernel of the linearized operator, the search space $X$ is restricted to functions satisfying:
     • Evenness in $x_2, x_3, x_4$.
     • Rotational invariance of $2\pi/k$ in the $(x_1, x_2)$ plane.
     • Invariance under the Kelvin transform (crucial for the critical case in $\mathbb{R}^4$).

3. Finite-Dimensional Reduction:

   • The system is rewritten as $L(\phi_1, \phi_2) = E + N(\phi_1, \phi_2)$, where $L$ is the linearized operator, $E$ is the error term from the approximation, and $N$ contains higher-order nonlinearities.
   • Invertibility: The author proves that the linear operator $L$ is invertible on the orthogonal complement of its kernel ($K^\perp$). This relies on the non-degeneracy of the single Choquard equation solutions (established in recent literature).
   • Error Estimates: Rigorous estimates are performed to bound the error term $E$ and the nonlinear remainder $N$, showing they are small when $|\beta|$ is small and $\lambda$ is large.

4. Reduced Problem:

   • The existence of a solution reduces to finding a parameter $\lambda$ such that the projection of the error onto the kernel vanishes (i.e., solving $c(\lambda) = 0$).
   • This leads to an asymptotic expansion of the reduced energy functional involving terms of order $\frac{1}{\lambda^2}$ and $\frac{\beta}{\lambda^2} \ln(\frac{1}{\lambda})$.

3. Key Contributions

• Bridging Theories: This work is the first to bridge the theory of competitive coupled systems with the critical Choquard equation. While both fields have been studied separately, their interaction in the critical regime was an open frontier.
• Handling Criticality in $\mathbb{R}^4$: The paper specifically addresses the dimension $N=4$. The author notes that for $N \ge 5$, the coupling term has sub-linear growth, which prevents the reduction method from working in the same way. The $N=4$ case requires delicate handling of the critical exponent and the Kelvin transform invariance.
• Segregation Mechanism: It demonstrates that even with a nonlocal interaction (which usually implies long-range effects), the repulsive force $\beta < 0$ is sufficient to drive the components to separate spatially. One component remains diffuse (ground state), while the other concentrates into a multi-bump structure.
• Explicit Asymptotics: The paper provides precise asymptotic formulas for the concentration scale $\lambda$ in terms of the coupling parameter $\beta$:
  $$\frac{1}{\lambda} \sim \exp\left( \frac{c}{|\beta|} \right)$$
  This indicates an exponential separation rate as the repulsion becomes weak.

4. Main Results

Theorem 1.2:
For every fixed integer $k \ge 2$, there exists a threshold $\beta_0 < 0$ such that for all $\beta \in [\beta_0, 0)$, the system admits a solution $(u, v)$ of the form:
$$u = U + \phi_1, \quad v = \sum_{j=1}^k U_{\lambda, \xi_j} + \phi_2$$
where:

• $U$ is the radial ground state of the single equation.
• $U_{\lambda, \xi_j}$ are bubbles centered at the vertices of a regular $k$-gon.
• $\phi_1, \phi_2$ are small remainder terms in the weighted space $D^{1,2}(\mathbb{R}^4)$.
• The concentration parameter satisfies $\frac{1}{\lambda} = O(e^{-c/|\beta|})$ as $|\beta| \to 0$.

5. Significance

• Mathematical Physics: The Choquard equation models physical phenomena such as polaron theory (electron-phonon interaction) and boson stars. Understanding coupled systems with repulsive forces is relevant for modeling multi-component quantum systems or plasma dynamics where different species repel each other.
• Analytical Technique: The paper refines the Lyapunov-Schmidt reduction technique for critical nonlocal problems. The detailed estimates involving the HLS kernel and the specific geometry of the regular polygon in $\mathbb{R}^4$ provide a template for future studies on similar critical systems.
• Segregation Phenomena: It confirms that the "segregation" phenomenon (where components of a system separate into disjoint regions) persists even in the presence of nonlocal critical nonlinearities, provided the interspecies repulsion is sufficiently strong (or the coupling is sufficiently small in magnitude but negative).

In summary, Caputo successfully constructs complex, multi-bump segregated solutions for a critical nonlocal system, proving that the interplay between critical Sobolev growth, nonlocal interactions, and competitive coupling allows for rich solution structures characterized by specific geometric arrangements of concentration points.