Instability of Standing Waves for the Nonlinear Schrodinger-Poisson Equation in the L2-Critical Case

In this paper, we consider the strong instability of standing waves for the nonlinear Schrodinger-Poisson equation i partial derivative(t)psi + Delta psi - (vertical bar x vertical bar(-1) * vertical bar psi vertical bar(2))psi + vertical bar psi vertical bar(p) psi = 0 (t, x) is an element of [0, T*) x R-3. In the L-2-critical case, i.e., p = 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., 4/3 < p < 4.