Blowing-up solutions to competitive critical systems in dimension 3

Abstract

We study the critical system of $m\ge 2$ equations

where ${\beta }_{\kappa \ell }=\alpha \in \mathbb{R}$ if $\kappa \ne \ell$, and ${\beta }_{\ell m}={\beta }_{m\kappa }=\beta <0$, for $\kappa ,\ell \in \{1,\dots ,m-1\}$. We construct solutions to this system in the case where $\beta \to -\infty$ by means of a Ljapunov–Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: $u_1$ will look like a perturbation of the standard radial positive solution to the Yamabe equation, while $u_2$ will blow-up at the $k$ vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures $u_i\ne u_j$ for $i\ne j$. The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3.