Differential and Integral Equations

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

Pierre-Damien Thizy

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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]).

Article information

Differential Integral Equations, Volume 30, Number 11/12 (2017), 825-832.

First available in Project Euclid: 1 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 35J47: Second-order elliptic systems 35Q61: Maxwell equations 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx] 58J37: Perturbations; asymptotics 81Q35: Quantum mechanics on special spaces: manifolds, fractals, graphs, etc.


Thizy, Pierre-Damien. Unstable phases for the critical Schrödinger-Poisson system in dimension 4. Differential Integral Equations 30 (2017), no. 11/12, 825--832. https://projecteuclid.org/euclid.die/1504231275

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