Blowing-up solutions for the Yamabe equation

Abstract

Let $(M,g)$ be a smooth, compact Riemannian manifold of dimension $N\ge 3$. We consider the almost critical problem

where ${\Delta }_g$ denotes the Laplace-Beltrami operator, ${\mathrm{Scal}}_g$ is the scalar curvature of $g$ and $ϵ\in \mathbb{R}$ is a small parameter. It is known that problem $(P_{ϵ})$ does not have any blowing-up solutions when $ϵ\nearrow 0$ , at least for $N\le 24$ or in the locally conformally flat case, and this is not true anymore when $ϵ\searrow 0$ . Indeed, we prove that, if $N\ge 7$ and the manifold is not locally conformally flat, then problem $(P_{ϵ})$ does have a family of solutions which blow-up at a maximum point of the function $\xi \to {\left|{\mathrm{Weyl}}_g(\xi )\right|}_g$ as $ϵ\searrow 0.$ Here Weyl${}_g$ denotes the Weyl curvature tensor of $g.$