Blowing-up solutions to a critical 4D Neumann system in a competitive regime

1. Problem Statement

The paper investigates the existence of boundary blow-up solutions for a critical elliptic system with Neumann boundary conditions in a four-dimensional bounded domain $\Omega \subset \mathbb{R}^4$. The system is given by:

$$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2 & \text{in } \Omega, \\ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_2 u_1^2 & \text{in } \Omega, \\ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0 & \text{on } \partial \Omega, \end{cases}$$

Key Parameters and Conditions:

• Dimension: $N=4$ (Critical Sobolev exponent $2^* = \frac{2N}{N-2} = 4$).
• Parameter $\lambda$: A large positive parameter ($\lambda \to +\infty$).
• Coupling Constant $\beta$: Positive ($\beta > 0$), representing a competitive regime (repulsive interaction between components).
• Domain Geometry: $\Omega$ is a smooth bounded domain with specific geometric features (protrusions) that influence the mean curvature.

The primary goal is to construct solutions $(u_{1,\lambda}, u_{2,\lambda})$ where each component $u_i$ concentrates (blows up) at distinct points $\xi_i$ on the boundary $\partial \Omega$ as $\lambda \to \infty$.

2. Methodology: Lyapunov–Schmidt Reduction

The authors employ 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:

A. Construction of the Ansatz

The solution is constructed in the form $u_i = V_{\lambda, \delta_i, \xi_i} + \psi_i$, where:

• Main Term ($V$): A superposition of a "bubble" and a correction term.
  • $U_{\delta, \xi}$ is the standard Talenti bubble solving the critical problem in $\mathbb{R}^4$.
  • $W_{\lambda, \delta, \xi}$ is a correction term introduced by Rey (adapted for $N=4$) involving Bessel functions. This term is crucial because the standard bubble does not satisfy the Neumann boundary condition or the specific asymptotic behavior required in dimension 4.
  • The ansatz is defined as $V_{\lambda, \delta, \xi} = U_{\delta, \xi} - W_{\lambda, \delta, \xi}$.
• Remainder ($\psi$): A small error term orthogonal to the kernel of the linearized operator.

B. Finite-Dimensional Reduction

1. Linearization: The problem is projected onto the orthogonal complement of the kernel space $K^\perp$ and the kernel space $K$ (spanned by derivatives of the bubble with respect to scaling and tangential parameters).
2. Solving the Orthogonal Part: Using the contraction mapping principle, the authors prove the existence of a unique small remainder $\psi$ for any given concentration parameters $(\delta, \xi)$, provided $\lambda$ is sufficiently large. This reduces the problem to finding critical points of a reduced energy functional.
3. Reduced Energy Expansion: The energy functional $\mathcal{E}$ is expanded in terms of the parameters $\delta_i$ (concentration scales) and $\xi_i$ (concentration points). The expansion reveals that the leading order terms depend on the mean curvature $H(\xi)$ of the boundary.

3. Key Contributions and Technical Innovations

• Handling the Critical Dimension $N=4$:
  Unlike higher dimensions ($N \ge 6$) where the standard Talenti bubble suffices, the case $N=4$ is delicate due to logarithmic divergences and the specific decay rates of the Green's function. The authors introduce a modified ansatz incorporating Bessel functions ($K_1$) to accurately capture the boundary behavior and higher-order terms in the expansion.
• Competitive Regime Analysis:
  The paper addresses the competitive case ($\beta > 0$). In this regime, the interaction term $-\beta u_1 u_2^2$ forces the components to separate. The authors show that this separation allows for the construction of solutions where $u_1$ and $u_2$ blow up at distinct boundary points, rather than collapsing into a single point.
• Geometric Dependence:
  The existence of solutions is strictly tied to the geometry of the domain. The reduced energy functional shows that the concentration points $\xi_i$ must be strict local maxima of the mean curvature $H$ with $H(\xi_i) > 0$.

4. Main Results

Theorem 1.1 (Main Existence Result):
Assume the mean curvature function $H$ of $\partial \Omega$ admits two distinct strict local maximum points $\xi_1^*$ and $\xi_2^*$ such that $H(\xi_i^*) > 0$. Then, for any $\beta > 0$, there exists a threshold $\lambda_0 > 0$ such that for all $\lambda > \lambda_0$, the system admits a solution $(u_{1,\lambda}, u_{2,\lambda})$ where:

• $u_{1,\lambda}$ blows up at $\xi_1^*$.
• $u_{2,\lambda}$ blows up at $\xi_2^*$.
• The concentration scales behave as $\delta_i \sim \frac{d_i}{\lambda \ln \lambda}$.

Corollaries and Remarks:

• Ellipsoids: Since an ellipsoid has at least two strict local maxima of positive mean curvature, the system always admits blowing-up solutions in such domains.
• Multiple Protrusions: If the domain has $k$ protrusions (tips with positive local mean curvature), the system admits $\frac{k(k-1)}{2}$ distinct blowing-up solutions for large $\lambda$.
• Generalization: The results extend to systems with $d \ge 2$ components in a fully competitive regime.

5. Significance and Impact

• Filling a Gap in Literature: Prior to this work, the existence of boundary blow-up solutions for critical Neumann systems in the competitive regime was an open problem, especially in low dimensions ($N=3, 4, 5$). This paper provides the first rigorous construction for $N=4$.
• Refinement of Techniques: The work demonstrates that the standard "bubble" ansatz is insufficient for $N=4$ Neumann problems. The successful integration of Bessel-function-based corrections sets a precedent for analyzing critical problems in lower dimensions where compactness is lost and boundary effects are dominant.
• Physical Relevance: The system models coupled Schrödinger equations found in nonlinear optics and Bose-Einstein condensates. Understanding how competitive interactions lead to spatial segregation (phase separation) at the boundary in the critical regime is vital for these physical applications.
• Geometric Control: The result highlights the profound influence of domain geometry (specifically mean curvature) on the formation of singularities in competitive systems, offering a tool to predict solution behavior based on domain shape.

In summary, this paper successfully bridges the gap between scalar critical Neumann problems and coupled systems in the critical dimension $N=4$, utilizing advanced asymptotic analysis and geometric insights to prove the existence of multi-peak boundary solutions in a competitive setting.