# Optimality conditions for optimal control of multisolution p-Laplacian elliptic equations

## TL;DR

This paper establishes optimality conditions for control problems governed by p-Laplacian elliptic equations with multiple solutions, addressing challenges from non-monotonic nonlinearities and solution multiplicity through penalization and approximation techniques.

## Contribution

It introduces a novel approach to derive optimality conditions without monotonicity assumptions, handling solution multiplicity and degeneracy in p-Laplacian control problems.

## Key findings

- Derived optimality conditions for multisolution p-Laplacian control problems.
- Developed penalization and approximation methods to handle degeneracy.
- Proved main results via limit processes.

## Abstract

In this paper, an optimal control problem governed by a class of p-Laplacian elliptic equations is studied. In particular, as no monotonicity assumption is assumed on the nonlinear term, the state equation may admit several solutions for one control. To obtain optimality conditions for an optimal pair, the multiplicity and singularity/degeneracy of the state equation need to be handled respectively. For this reason, penalization problems and approximation problems are introduced. Finally the main result is proved by a series of process of taking to the limits.