Thizy, Pierre-Damien Unstable phases for the critical Schrödinger-Poisson system in dimension 4. (English) Zbl 1413.58009 Differ. Integral Equ. 30, No. 11-12, 825-832 (2017). Summary: We consider, in this note, the critical Schrödinger-Poisson system \[ \begin{cases}\Delta_gu+\omega^2u+\varphi u &=u^{\frac{n+2}{n-2}},\\ \Delta_g\varphi +m_0^2\varphi &=4\pi q^2u^2\end{cases}\tag{0.1} \] 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 [the author, Discrete Contin. Dyn. Syst. 36, No. 4, 2257–2284 (2016; Zbl 1328.58016)]) when \(n=4\) and the scalar curvature is positive everywhere, and unstable phases never exist when \(n\geq 5\) (see the author, Arch. Math. 104, No. 5, 485–490 (2015; Zbl 1320.58015); “Phase stability for Schrödinger-Poisson critical systems in closed \(5\)-manifolds”, Int. Math. Res. Not. 2016, No. 20, 6245–6292 (2016)]). Cited in 3 Documents MSC: 58J05 Elliptic equations on manifolds, general theory 35J47 Second-order elliptic systems 35Q61 Maxwell equations 35R01 PDEs on manifolds 58J37 Perturbations of PDEs on manifolds; asymptotics 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 35B09 Positive solutions to PDEs Keywords:unstable phase; positive solution; existence; standing waves solution Citations:Zbl 1328.58016; Zbl 1320.58015 × Cite Format Result Cite Review PDF