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

Abstract

Abstract We prove the existence of two smooth families of unbounded domains in ℝ N + 1 {\mathbb{R}^{N+1}} with N ≥ 1 {N\geq 1} such that { - Δ ⁢ u = λ ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , ∂ ν ⁡ u = const on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta u&\displaystyle=\lambda u&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\\ \displaystyle\partial_{\nu}u&\displaystyle=\mathrm{const}&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. admits a sign-changing solution. The domains bifurcate from the straight cylinder B 1 × ℝ {B_{1}\times\mathbb{R}} , where B 1 {B_{1}} is the unit ball in ℝ N {\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.