  We show that a wide range of overdetermined boundary problems for semilinear
equations with position-dependent nonlinearities admits nontrivial solutions.
The result holds true both on the Euclidean space and on compact Riemannian
manifolds. As a byproduct of the proofs we also obtain some rigidity, or
partial symmetry, results for solutions to overdetermined problems on
Riemannian manifolds of nonconstant curvature.
