  In this paper we prove existence of normalized solutions $(u,\lambda)\subset
H^1(\mathbb{R}^3)\times \mathbb{R}^+$ of the following Schr\"{o}dinger-Poisson
equation \begin{equation} \begin{aligned} -\Delta u+V(x)u+\lambda
u+(|x|^{-1}\ast|u|^2)u&=|u|^{p-2}u,&\text{in}~\mathbb{R}^{3},\\
\int_{\mathbb{R}^{3}}u^2dx&=a^2>0, \end{aligned} \end{equation} under suitable
conditions on the potential $V(x)$, where $\lambda$ appears as a Lagrange
multiplier. For $V(x)\geq0$ and $p\in(4,6)$, such solutions are obtained by
using a linking argument introduced by Bartsch et al. \cite{bm}. For
$V(x)\leq0$, a mountain pass argument leads to the existence of normalized
solutions for any $p\in(\frac{10}{3},6)$.

Analysis of PDEs