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

Abstract

In 1971 J. Serrin proved that, given a smooth bounded domain $\Omega \subset {\mathbb{R}}^N$ and a positive solution $u$ of the problem

$\Omega$ 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.