  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.
