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

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 $\lambda >0$ is sufficiently large in a competitive regime (i.e. $\beta >0$) and in a domain $\mathrm{\Omega }\subset {\mathbb{R}}^4$ with smooth protrusions.