  The paper deals with two nonlinear elliptic equations with $(p,q)$-Laplacian
and the Dirichlet-Neumann-Dirichlet (DND) boundary conditions, and
Dirich\-let-Neu\-mann-Neumann (DNN) boundary conditions, respectively. Under
mild hypotheses, we prove the unique weak solvability of the elliptic mixed
boundary value problems. Then, a comparison and a monotonicity results for the
solutions of elliptic mixed boundary value problems are established. We examine
a convergence result which shows that the solution of (DND) can be approached
by the solution of (DNN). Moreover, two optimal control problems governed by
(DND) and (DNN), respectively, are considered, and an existence result for
optimal control problems is obtained. Finally, we provide a result on
asymptotic behavior of optimal controls and system states, when a parameter
tends to infinity.
