  This work is concerned with an optimal control problem governed by a
non-smooth quasilinear elliptic equation with a nonlinear coefficient in the
principal part that is locally Lipschitz continuous and directionally but not
G\^ateaux differentiable. This leads to a control-to-state operator that is
directionally but not G\^ateaux differentiable as well. Based on a suitable
regularization scheme, we derive C- and strong stationarity conditions. Under
the additional assumption that the nonlinearity is a PC^1 function with
countably many points of nondifferentiability, we show that both conditions are
equivalent. Furthermore, under this assumption we derive a relaxed optimality
system that is amenable to numerical solution using a semi-smooth Newton
method. This is illustrated by numerical examples.
