  We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$
  - \Delta u + (I_\alpha \ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in
\(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is
the Riesz potential of order $\alpha\in(0,N).$ We introduce and study the
Coulomb-Sobolev function space which is natural for the energy functional of
the problem and we establish a family of associated optimal interpolation
inequalities. We prove existence of optimizers for the inequalities, which
implies the existence of solutions to the equation for a certain range of the
parameters. We also study regularity and some qualitative properties of
solutions. Finally, we derive radial Strauss type estimates and use them to
prove the existence of radial solutions to the equation in a range of
parameters which is in general wider than the range of existence parameters
obtained via interpolation inequalities.
