  In this paper we consider the overdetermined boundary problem for a general
second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$,
where one prescribes both the Dirichlet and Neumann data of the solution. We
are interested in the case where the data are not necessarily constant and
where the coefficients of the equation can depend on the position, so that the
overdetermined problem does not generally admit a radial solution. Our main
result is that, nevertheless, under minor technical hypotheses nontrivial
solutions to the overdetermined boundary problem always exist.
