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

Overdetermined boundary problems with nonconstant Dirichlet and Neumann data