Multiple normalized solutions for a Sobolev critical Schr\"{o}dinger-Poisson-Slater equation

We look for solutions to the Schr"{o}dinger-Poisson-Slater equation $$-\mathrm{\Delta }u+\lambda u-\gamma (\mathrm{\mid }x{\mathrm{\mid }}^{-1}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u-a\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=0\text{in}{\mathbb{R}}^3,$$ which satisfy \begin{equation*} \int_{$\mathbb{R}$^3}|u|^2 , dx = c \end{equation*} for some prescribed $c>0$. Here $u\in H^1({\mathbb{R}}^3)$, $\gamma \in \mathbb{R},$ $a\in \mathbb{R}$ and $p\in (\frac{10}{3},6]$. When $\gamma >0$ and $a>0$, both in the Sobolev subcritical case $p\in (\frac{10}{3},6)$ and in the Sobolev critical case $p=6$, we show that there exists a $c_1>0$ such that, for any $c \in (0,c_1)$,theequationadmitstwosolutions$u_c^+$and$u_c^-$ which can be characterized respectively as a local minima and as a mountain pass critical point of the associated {\it Energy} functional restricted to the norm constraint. In the case $\gamma >0$ and $a<0$, we show that, for any $p \in (\frac{10}{3},6]$andany$c>0$, the equation admits a solution which is a global minimizer. Finally, in the case $\gamma <0$, $a>0$ and $p=6$ we show that it does not admit positive solutions.