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

ZENG, SHENGDA; MIGÓRSKI, STANIS?AW; TARZIA, DOMINGO A.; ZOU, LANG; NGUYEN, VAN THIEN

ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK

BIRKHAUSER VERLAG AG

Año: 2022 vol. 73

0044-2275

The paper deals with two nonlinear elliptic equations with (p, q)-Laplacian and the Dirichlet–Neumann–Dirichlet (DND) boundary conditions, and Dirichlet–Neumann–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.