Bifurcation domains from any high eigenvalue for an overdetermined elliptic problem

In this paper, we consider the following overdetermined eigenvalue problem on an unbounded domain $\Omega\subset\mathbb{R}^{N+1}$ with $N\geq1$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u=\lambda u\,\, &\text{in}\,\, \Omega,\\ u=0 &\text{on}\,\, \partial \Omega,\\ \partial_\nu u=\text{const} &\text{on}\,\, \partial \Omega. \end{array} \right.\nonumber \end{equation} Let $\lambda_k$ be the $k$-th eigenvalue of the zero-Dirichlet Laplacian on the unit ball for any $k\in \mathbb{N^+}$ with $k\geq 3$. We can construct $k$ smooth families of nontrivial unbounded domains $\Omega$, bifurcating from the straight cylinder, which admit a nonsymmetric solution with changing the sign by $k-1$ times to the overdetermined problem. While the existence of such domains for $k=1,2$ has been well-known, to the best of our knowledge this is the first construction for any positive integer $k\geq 3$. Due to the complexity of studying high eigenvalue problem, our proof involves some novel analytic ingredients. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain.
Comment: 33 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:2304.05550