  This paper is concerned with an optimal control problem governed by a
non-smooth semilinear elliptic equation. We show that the control-to-state
mapping is directionally differentiable and precisely characterize its
Bouligand subdifferential. By means of a suitable regularization, first-order
optimality conditions including an adjoint equation are derived and afterwards
interpreted in light of the previously obtained characterization. In addition,
the directional derivative of the control-to-state mapping is used to establish
strong stationarity conditions. While the latter conditions are shown to be
stronger, we demonstrate by numerical examples that the former conditions are
amenable to numerical solution using a semi-smooth Newton method.
