  We consider the inverse problem of determining a potential in a semilinear
elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For
bounded Euclidean domains we prove that the potential is uniquely determined by
the Dirichlet-to-Neumann map measured at a single boundary point, or integrated
against a fixed measure. This result is valid even when the Dirichlet data is
only given on a small subset of the boundary. We also give related uniqueness
results on Riemannian manifolds.
