Stability and instability results for sign-changing solutions to second-order critical elliptic equations

Abstract

On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge 3$, we consider the stationary Schrödinger equation ${\Delta }_{g}u+h_{0}u={\left|u\right|}^{2^{\ast }-2}u$, where ${\Delta }_g:=-{\text{div}}_g\nabla$, $h_0\in C^1\left(M\right)$ and $2^{\ast }:=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential function $h_0$ in $C^1\left(M\right)$, the sets of sign-changing solutions that are bounded in $H^1\left(M\right)$ are precompact in the $C^2$ topology. We obtain this result under the assumptions that $\left(M,g\right)$ is locally conformally flat, $n\ge 7$ and $h_0\ne \frac{n-2}{4\left(n-1\right)}{\text{Scal}}_g$ at all points in $M$, where ${\text{Scal}}_g$ is the scalar curvature of the manifold. We then provide counterexamples in every dimension $n\ge 3$ showing the optimality of these assumptions.

Final version, published in Journal de Mathématiques Pures et Appliquées