Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations

In this paper we study the existence and the strong instability of standing waves with prescribed $L^2$-norm for a class of Schr\"odinger-Poisson-Slater equations in $\R^{3}$ \label{evolution1} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 %\ \ \text{in} \R^{3}, when $10/3<p0$ is sufficiently small. We show for initial condition $u_0\in H^1(\R^3)$ of the associated Cauchy problem with $\|u_0\|_{2}^2=c$ that the mountain pass energy level $\gamma(c)$ gives a threshold for global existence. The strong instability of standing waves at mountain pass energy levels is proved adapting the Berestycki-Cazenave's approach. We also draw a comparison between the Schr\"odinger-Poisson-Slater equation and the classical nonlinear Schr\"odinger equation.