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

Abstract

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.

32 pages, 3 figures