Sign-changing solutions to overdetermined elliptic problems in bounded domains

Abstract.

In 1971 J. Serrin proved that, given a smooth bounded domain Ω ⊂ RN and u a positive solution of the problem:

−∆u = f(u) in Ω, u = 0 on ∂Ω, ∂ν u = constant on ∂Ω,

then \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 C2 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.