  In this work, we consider an optimal control problem subject to a nonlinear
PDE constraint and apply it to the regularized $p$-Laplace equation. To this
end, a reduced unconstrained optimization problem in terms of the control
variable is formulated. Based on the reduced approach, we then derive an a
posteriori error representation and mesh adaptivity for multiple quantities of
interest. All quantities are combined to one, and then the dual-weighted
residual (DWR) method is applied to this combined functional. Furthermore, the
estimator allows for balancing the discretization error and the nonlinear
iteration error. These developments allow us to formulate an adaptive solution
strategy, which is finally substantiated via several numerical examples.
