  We exhibit several counterexamples showing that the famous Serrin's symmetry
result for semilinear elliptic overdetermined problems may not hold for
partially overdetermined problems, that is when both Dirichlet and Neumann
boundary conditions are prescribed only on part of the boundary. Our
counterexamples enlighten subsequent positive symmetry results obtained by the
first two authors for such partially overdetermined systems and justify their
assumptions as well.
