Unstable phases for the critical Schrödinger-Poisson system in dimension 4

Abstract

We consider, in this note, the critical Schrödinger-Poisson system \begin{equation}\label{SP0} \begin{cases} \Delta_g u+ \omega^2 u +\varphi u = u^{\frac{n+2}{n-2}}~,\\ \Delta_g \varphi +m_0^2 \varphi = 4\pi q^2 u^2~ \end{cases} \end{equation} on a closed Riemannian $n$-dimensional manifold $(M^n,g)$, for $n=4$. If the scalar curvature is negative somewhere, we prove that this system admits positive solutions for small phases $\omega$ and that $\omega=0$ is an unstable phase (see Definition 1.1. By contrast, small phases are always stable (see [32]) when $n=4$ and the scalar curvature is positive everywhere, and unstable phases never exist when $n\ge 5$ (see [29, 31]).