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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.

Counterexamples to symmetry for partially overdetermined elliptic problems