On uniqueness and radiality of minimizers to $L^2$ supercritical Schrödinger Poisson equations with general nonlinearities

We study the uniqueness and the radial symmetry of minimizers on a Pohozaev-Nehari manifold to the Schrödinger Poisson equation with a general nonlinearity $f(u)$. Particularly, we allow that $f$ is $L^2$ supercritical. The main result shows that minimizers are unique and radially symmetric modulo suitable translations.