# Blowing-up solutions for the Yamabe equation

### Pierpaolo Esposito

Università degli Studi "Roma Tre", Italy### Angela Pistoia

Università di Roma "La Sapienza", Italy

## Abstract

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

where $Δ_{g}$ denotes the Laplace-Beltrami operator, $Scal_{g}$ is the scalar curvature of $g$ and $ϵ∈R$ is a small parameter. It is known that problem $(P_{ϵ})$ does not have any blowing-up solutions when $ϵ↗0$ , at least for $N≤24$ or in the locally conformally flat case, and this is not true anymore when $ϵ↘0$ . Indeed, we prove that, if $N≥7$ 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 $ξ→∣∣ Weyl_{g}(ξ)∣∣ _{g}$ as $ϵ↘0.$ Here Weyl$_{g}$ denotes the Weyl curvature tensor of $g.$

## Cite this article

Pierpaolo Esposito, Angela Pistoia, Blowing-up solutions for the Yamabe equation. Port. Math. 71 (2014), no. 3/4, pp. 249–276

DOI 10.4171/PM/1952