  In this paper, we investigate optimal control problems subject to a
semilinear elliptic partial differential equations. The cost functional
contains a term that measures the size of the support of the control, which is
the so-called $L^0$-norm. We provide necessary and sufficient optimality
conditions of second-order. The sufficient second-order condition is obtained
by analyzing a partially convexified problem. Interestingly, the structure of
the problem yields second-order conditions with different bilinear forms for
the necessary and for the sufficient condition.
