Instability of Standing Waves for the Nonlinear Schrödinger–Poisson Equation in the $$L^2$$-Critical Case

Abstract

In this paper, we consider the strong instability of standing waves for the nonlinear Schrodinger–Poisson equation $$\begin{aligned} i\partial _t\psi +\Delta \psi -(|x|^{-1}*|\psi |^2)\psi +|\psi |^{p}\psi =0~~~~ (t,x)\in [0,T^*)\times {\mathbb {R}}^3. \end{aligned}$$ In the $$L^2$$ -critical case, i.e., $$p=\frac{4}{3}$$ , we prove that the standing waves are strongly unstable by blow-up. This result is a complement to the result of Kikuchi (Adv Nonlinear Stud 7:403–437, 2007) and Bellazzini et al. (Proc Lond Math Soc 107:303–339, 2013), where the instability of standing waves were studied in the $$L^2$$ -supercritical case, i.e., $$\frac{4}{3}< p<4$$ .