  An optimal control problem for a semilinear elliptic equation of divergence
form is considered. Both the leading term and the semilinear term of the state
equation contain the control. The well-known Pontryagin type maximum principle
for the optimal controls is the first-order necessary condition. When such a
first-order necessary condition is singular in some sense, certain type of the
second-order necessary condition will come in naturally. The aim of this paper
is to explore such kind of conditions for our optimal control problem.
