Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds. II (Q274660)

This paper concerns stability of standing wave solutions of the static Klein-Gordon-Maxwell-Proca system in 4-dimensional closed manifold. Writing the matter scalar field in the form \(\psi(x,t)=u(x)e^{-i\omega t}\), the system is reduced to NEWLINE\[NEWLINE\begin{cases} \Delta_g u+m_0^2u=u^3+\omega^2(qv-1)^2u,\\ \Delta_g v+(m_1^2+q^2u)v=qu^2. \end{cases} NEWLINE\]NEWLINE The notation of phase stability is proposed, and the condition for phase \(\omega\) to be stable is given.