  We study the existence of nontrivial unbounded domains $\Omega$ in
$\mathbb{R}^N$ such that the overdetermined problem $$ -\Delta u = 1 \quad
\text{in $\Omega$}, \qquad u=0, \quad \partial_\nu u=\textrm{const} \qquad
\text{on $\partial \Omega$} $$ admits a solution $u$. By this, we complement
Serrin's classification result from 1971 which yields that every bounded domain
admitting a solution of the above problem is a ball in $\mathbb{R}^N$. The
domains we construct are periodic in some variables and radial in the other
variables, and they bifurcate from a straight (generalized) cylinder or slab.
We also show that these domains are uniquely self Cheeger relative to a period
cell for the problem.
