Sign-changing solution for an overdetermined elliptic problem on unbounded domain

We prove the existence of two smooth families of unbounded domains in ${\mathbb{R}}^{N+1}$ with $N\ge 1$ such that \begin{equation} -Δu=λu,, $\text{in}$,,Ω, ,, u=0,,,\partial_νu=$\text{const}$,,$\text{on}$,,\partialΩ\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.