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

Abstract: We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -Δu_1 + λu_1 = u_1^{3} -βu_1u_2^2 & \text{in } Ω, -Δu_2 + λu_2 = u_2^{3} -βu_1^2u_2 & \text{in } Ω, \frac{\partial u_1}{\partialν} = \frac{\partial u_2}{\partialν} = 0, & \text{on } \partial Ω, \end{cases} \end{equation*} when $λ>0$ is sufficiently large in a competitive regime (i.e. $ β>0$) and in a domain $Ω\subset\mathbb R^4$ with smooth protrusions.

• The paper establishes the existence of blow-up solutions to a critical 4D Neumann system under competitive regimes.
• It applies a Lyapunov-Schmidt reduction and refined ansatz incorporating Bessel function corrections to control concentration effects.
• Results quantify error bounds and highlight the pivotal role of boundary curvature in determining blow-up locations.

Blowing-up Solutions for Critical 4D Neumann Systems in the Competitive Regime

Problem Formulation and Context

The paper addresses the existence and construction of blowing-up solutions for the following critical elliptic Neumann system in a smooth bounded domain $\Omega\subset\mathbb{R}^4$: $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$ with $\lambda > 0$ large and $\beta > 0$ (strictly competitive regime). The Sobolev critical exponent and boundary conditions pose significant analytical challenges due to loss of compactness and the strong interaction between nonlinearity and domain geometry.

This coupled model is motivated by its emergence in interacting Schrödinger systems, relevant for multi-component Bose-Einstein condensates and nonlinear optics. The mathematical structure introduces additional complexity relative to the scalar critical Neumann problem, particularly in four dimensions, where analytic blow-up constructions require fine asymptotic control.

Methodological Framework

The primary analytical tool is a finely tuned Lyapunov-Schmidt reduction, extending techniques from the scalar critical Neumann problem [Wei-Yan, J. Math. Pures Appl. 88 (2007)], [Rey, Commun. Contemp. Math. 1 (1999)]. The solution ansatz for each component is

$u_i = V_{\lambda,\delta_i,\xi_i} + \psi_i,$

where the primary term incorporates a boundary-concentrated bubble $U_{\delta_i,\xi_i}$, corrected by $W_{\lambda,\delta_i,\xi_i}$ (a refined Bessel-type adjustment necessary for $N=4$), with $\delta_i = \frac{d_i}{\lambda \ln\lambda}$ controlling concentration, and $\xi_i \in \partial\Omega$ denoting the blow-up location.

The reduction decouples the infinite-dimensional perturbative problem (solved for $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$0 in the orthogonal complement of an approximate kernel) from a finite-dimensional problem in $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$1, with solution existence linked to critical points of an explicitly derived reduced energy functional.

A major technical obstacle arises in the necessity for boundary layers and the non-adequacy of Talenti bubbles in 4D. The authors develop a highly regular ansatz involving Bessel functions to capture higher-order effects in the blow-up regime.

Core Results

The main theorem demonstrates that for any $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$2, provided that the mean curvature $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$3 of $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$4 admits $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$5 distinct strict local maxima with $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$6, the system above has solutions $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$7 such that $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$8 blows up at $$\begin{cases} -\Delta u_1 + \lambda u_1 = u_1^3 - \beta u_1 u_2^2, & \text{in } \Omega, \ -\Delta u_2 + \lambda u_2 = u_2^3 - \beta u_1^2 u_2, & \text{in } \Omega, \ \frac{\partial u_1}{\partial \nu} = \frac{\partial u_2}{\partial \nu} = 0, & \text{on } \partial\Omega, \end{cases}$$9 as $\lambda > 0$0. This extends to $\lambda > 0$1-component systems in the fully competitive case with concentration at distinct boundary points.

Quantitative Estimate: The error in the Lyapunov-Schmidt scheme is controlled via

$\lambda > 0$2

ensuring that the correction is lower order compared to the leading bubble.

Reduced Energy Expansion: The critical points of the reduced energy, whose expansion is

$\lambda > 0$3

select the concentration parameters and blow-up locations. The presence of local maxima of $\lambda > 0$4 guarantees the existence of isolated solutions.

Multiplicity: Domains with $\lambda > 0$5 strict local maxima for $\lambda > 0$6 admit at least $\lambda > 0$7 distinct blow-up solutions for $\lambda > 0$8, and more generally $\lambda > 0$9 for $\beta > 0$0-component systems.

Analytical and Geometric Insights

• The analysis foregrounds the crucial role of boundary geometry; blow-up is tightly localized at strict local maxima of curvature.
• The necessity of a refined ansatz (with Bessel function corrections) for $\beta > 0$1 marks a sharp contrast with higher-dimensional situations, where standard bubble profiles suffice.
• The existence and localization of solutions are dictated by the precise geometric structure of $\beta > 0$2 via the non-degeneracy of critical points of $\beta > 0$3.
• The procedure provides a concrete mechanism for symmetry-breaking and pattern selection in multi-component competitive systems, with potential implications for the study of heteroclinic structures and segregation phenomena.

Implications and Future Directions

The construction provides a comprehensive answer to the existence of boundary blow-up solutions for critical 4D Neumann systems in the competitive regime, filling a gap unaddressed in earlier works (2603.29329). The approach handles delicate regularity and nonlinearity interplay inherent to elliptic critical systems.

Potential Extensions:

• Generalization to degenerate or non-isolated boundary curvature critical points, which would require expansion of the reduced energy at a $\beta > 0$4 level or higher.
• Analysis of the asymptotic behavior/energy of least energy solutions in both the cooperative and competitive regimes, particularly their blow-up and concentration set geometry as $\beta > 0$5 [arxiv-2025].
• Extension to mixed or sign-changing regimes, which may necessitate different variational and reduction techniques.
• Investigation of stability and qualitative properties (e.g., Morse index, uniqueness) of the constructed solutions under perturbation of domain geometry, coupling coefficients, or boundary conditions.

Broader Applications: The methodology is relevant for multi-component reaction-diffusion systems in mathematical physics, notably in studies of pattern formation (cf. Gierer-Meinhardt models), competition-diffusion systems in population dynamics, and nonlinear optics. The highlights on curvature-driven multiplicity and localization have ramifications for design and control of phenomena involving interface localization in applied systems.

Conclusion

This work establishes the existence and characterization of boundary blow-up solutions for the 4D critical Neumann system in a strictly competitive regime, elucidating the geometric and analytic mechanisms underlying concentration in the presence of critical nonlinearities, and offering a refined reduction scheme that can serve as a template for further investigations into multi-component, critical Sobolev-type systems (2603.29329).