In 1971 J. Serrin proved that, given a smooth bounded domain Ω⊂Rn and u a positive solution of the problem: −Δu=f(u) in Ω, u=0 on ∂Ω, ∂_νu= constant on ∂Ω, then Ω 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 Ω different from a ball. The proof uses a local bifurcation argument, based on the study of the associated linearized operator. We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. In addition, rather surprisingly and differently from what happens for a scalar equation, the boundedness may fail if the exponents are not comparable.