Nonsymmetric sign-changing solutions to overdetermined elliptic problems in bounded domains

In 1971 J. Serrin proved that, given a smooth bounded domain \Omega \subset \mathbb{R}^{N} and a positive solution u of the problem \begin{cases} -\Delta u = f(u)&\text{in } \Omega, \\ u =0&\text{on } \partial\Omega, \\ \partial_{\nu}u =\text{constant}&\text{on }\partial\Omega \end{cases} \Omega is necessarily a ball and u is radially symmetric. In this paper we prove that the positivity of u is necessary in that symmetry result. In fact, we find a sign-changing solution to that problem for a C^{2} function f(u) in a bounded domain \Omega different from a ball. The proof uses a local bifurcation argument, based on the study of the associated linearized operator.