On existence, uniqueness and radiality of normalized solutions to Schrödinger-Poisson equations with non-autonomous nonlinearity

We investigate the existence, uniqueness, and radial symmetry of normalized solutions to the Schrödinger Poisson equation with non-autonomous nonlinearity $f(x,u)$: \begin{equation} -\triangle u+(|x|^{-1}*|u|^2)u=f(x,u)+λu, \nonumber \end{equation} subject to the constraint ${\mathcal{S}}_c=\{u\in H^1({\mathbb{R}}^3)\mathrm{\mid }{\int }_{{\mathbb{R}}^3}{u}^2=c>0\}$. We consider three cases based on the behavior of $f(x,u)$: the $L^2$ supercritical case, the $L^2$ subcritical case with growth speed less than three power times, and the $L^2$ subcritical case with growth speed more than three power times. We establish the existence of solutions using three different methods depending on $f(x,u)$. Furthermore, we demonstrate the uniqueness and radial symmetry of normalized solutions using an implicit function framework when $c$ is small.