Strong instability of standing waves for $L^2$-supercritical Schr\"odinger-Poisson system with a doping profile

This paper is devoted to the study of the nonlinear Schr\"odinger-Poisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the $L^2$-supercritical case. The presence of a doping profile causes several difficulties, especially in examining geometric shapes of fibering maps along an $L^2$-invariant scaling curve. Furthermore, the classical approach by Berestycki-Cazenave for the strong instability cannot be applied to our problem due to a remainder term caused by the doping profile. To overcome these difficulties, we establish a new energy inequality associated with the $L^2$-invariant scaling and adopt the strong instability result developed by Fukaya-Ohta(2018). When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the strong instability of standing waves.