Sign-changing blow-up for scalar curvature type equations

TL;DR

This paper constructs sign-changing solutions that blow up for scalar curvature equations on compact manifolds, revealing when solutions are compact or noncompact depending on dimension and potential.

Contribution

It demonstrates the existence of blowing-up sign-changing solutions in specific dimensions and potentials, expanding understanding of solution behavior for scalar curvature equations.

Findings

Existence of blow-up sign-changing solutions in dimensions 3-6 for any potential.

Existence of such solutions in dimensions 3-9 when potential equals a scalar curvature-related term.

Provides a complete picture of compactness and noncompactness for these equations.

Abstract

Given $(M,g)$ a compact Riemannian manifold of dimension $n\geq 3$, we are interested in the existence of blowing-up sign-changing families $(\ue)_{\eps>0}\in C^{2,\theta}(M)$, $\theta\in (0,1)$, of solutions to $$\Delta_g \ue+h\ue=|\ue|^{\frac{4}{n-2}-\eps}\ue\hbox{ in }M\,,$$ where $\Delta_g:=-\hbox{div}_g(\nabla)$ and $h\in C^{0,\theta}(M)$ is a potential. We prove that such families exist in two main cases: in small dimension $n\in \{3,4,5,6\}$ for any potential $h$ or in dimension $3\leq n\leq 9$ when $h\equiv\frac{n-2}{4(n-1)}\Scal_g$. These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet and Khuri--Marques--Schoen.