On optimality conditions for optimal control problem in coefficients for Δp-Laplacian

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as -coefficient of -Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincare Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair. MSC:35J70, 49J20, 49J45, 93C73