  We prove the existence of two smooth families of unbounded domains in
$\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda
u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu
u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a
sign-changing solution. The domains bifurcate from the straight cylinder
$B_1\times \mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These
results can be regarded as counterexamples to the Berenstein conjecture on
unbounded domain. Unlike most previous papers in this direction, a very
delicate issue here is that there may be two-dimensional kernel space at some
bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from
high-dimensional kernel space is also established to achieve the goal.
