In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{ R}^{ n+1},n\geq1$, 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} \Delta u +f(u)=0 &\mbox{ in $\Omega$, } u=0 &\mbox{ on $\partial\Omega$, } \partial_{ \nu} 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.
Categories: math.AP