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

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.