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

TL;DR

This paper constructs solutions that blow up for a critical elliptic Neumann system in four dimensions, focusing on large parameter regimes and domains with protrusions, revealing complex boundary behaviors.

Contribution

It introduces new blowing-up solutions for a critical 4D elliptic system with Neumann boundary conditions in a competitive regime, expanding understanding of boundary blow-up phenomena.

Findings

Constructed explicit blowing-up solutions in 4D domains with protrusions.

Analyzed the impact of large parameter $eta$ on solution behavior.

Identified boundary concentration phenomena in the solutions.

Abstract

We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -\Delta u_1 + \lambda u_1 = u_1^{3} -\beta u_1u_2^2 & \text{in } \Omega, -\Delta u_2 + \lambda u_2 = u_2^{3} -\beta u_1^2u_2 & \text{in } \Omega, \frac{\partial u_1}{\partial\nu} = \frac{\partial u_2}{\partial\nu} = 0, & \text{on } \partial \Omega, \end{cases} \end{equation*} when $\lambda>0$ is sufficiently large in a competitive regime (i.e. $ \beta>0$) and in a domain $\Omega\subset\mathbb R^4$ with smooth protrusions.