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

TL;DR

This paper investigates nonlinear elliptic equations with $(p,q)$-Laplacian under mixed boundary conditions, establishing existence, comparison, convergence, and optimal control results, including asymptotic behavior of solutions as parameters grow large.

Contribution

It introduces new existence and comparison results for mixed boundary value problems with $(p,q)$-Laplacian and analyzes optimal control problems and their asymptotic properties.

Findings

Unique weak solvability of the boundary value problems.

Solution convergence from (DNN) to (DND).

Existence of optimal controls and asymptotic behavior analysis.

Abstract

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.