  We prove the existence of nontrivial unbounded domains $\O$ in the Euclidean
space $\R^d$ for which the Dirichlet eigenvalue problem for the Laplacian on
$\Omega$ admits sign-changing eigenfunctions with constant Neumann values on
$\partial \Omega$. We also establish a similar result by studying a partially
overdetermined problem on domains with two boundary components and opposite
Neumann boundary values. The domains we construct are periodic in some
variables and radial in the other variables, and they bifurcate from straight
(generalized) cylinder or slab.
