Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities

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 R 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.The first author is supported by projects MTM2016-75897-P (AEI/FEDER) and ED431F 2017/03, by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 745722 , and by the Ramón y Cajal program of the Spanish Ministry of Science. The second and third authors have been supported by the ERC Starting Grants 633152 and 335079 , respectively. All authors acknowledge financial support from the Spanish Ministry of Science through the Severo Ochoa Program for Centers of Excellence in R&D ( SEV-2015-0554 ).Peer Reviewe