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

In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schrodinger-Poisson-Slater equations in $R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schrodinger-Poisson type equation label{evolution1} ipsi_{t}+ Delta psi - (|x|^{-1}*|psi|^{2}) psi+|psi|^{p-2}psi=0 % text{in} R^{3}, when $p in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| triangledown u|_{L^{2}(mathbb{R}^3)}^2+1/4int_{mathbb{R}^3}int_{mathbb{R}^3}frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-frac{1}{p}int_{mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= {u in H^1(mathbb{R}^3) :|u|_{L^2(R^3)}^2=c, c>0}.$$ For the values $p in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0in H^1(R^3)$ 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. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schrodinger-Poisson-Slater equation and the classical nonlinear Schrodinger equation.