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The paper deals with two nonlinear elliptic equations with $(p,q)$-Laplacianand the Dirichlet-Neumann-Dirichlet (DND) boundary conditions, andDirich\-let-Neu\-mann-Neumann (DNN) boundary conditions, respectively. Undermild hypotheses, we prove the unique weak solvability of the elliptic mixedboundary value problems. Then, a comparison and a monotonicity results for thesolutions of elliptic mixed boundary value problems are established. We examinea convergence result which shows that the solution of (DND) can be approachedby the solution of (DNN). Moreover, two optimal control problems governed by(DND) and (DNN), respectively, are considered, and an existence result foroptimal control problems is obtained. Finally, we provide a result onasymptotic behavior of optimal controls and system states, when a parametertends to infinity.

A class of elliptic mixed boundary value problems with $(p,q)$-Laplacian: existence, comparison and optimal control