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Unstable phases for the critical Schrödinger-Poisson system in dimension 4. (English) Zbl 1413.58009

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)]).

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