  We consider the Schr\"odinger-Poisson-Slater (SPS) system in $\R^3$ and a
nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary
conditions. We show that for every $k\in\mathbb N$ each problem considered
admits a nodal radially symmetric solution which changes sign exacly $k$ times
in the radial variable. Moreover when the domain is the ball of $\mathbb R^3$
we obtain the existence of radial global solutions for the associated nonlocal
parabolic problem having $k+1$ nodal regions at every time.
