Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

Abstract

In this paper, we prove the existence of nontrivial unbounded domains $\Omega \subset {\mathbb{R}}^{n+1},n\ge 1$, bifurcating from the straight cylinder $B\times \mathbb{R}$ (where $B$ is the unit ball of ${\mathbb{R}}^n$), such that the overdetermined elliptic problem \begin{equation*} \begin{cases} Δu +f(u)=0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial \Omega$, } \partial_ν u=\mbox{constant} &\mbox{on $\partial \Omega$, } \end{cases} \end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f:[0,+\infty )\to \mathbb{R}$. Roughly speaking, we only ask that the Dirichlet problem in $B$ admits a nondegenerate solution. The proof uses a local bifurcation argument.