Optimal potentials for problems with changing sign data

In this paper, we consider variational optimal control problems. The state equation is an elliptic partial differential equation of a Schr\"odinger type, governed by the Laplace operator with a potential, with a right-hand side that may change sign. The control variable is the potential itself, that may vary in a suitable admissible class of non-negative potentials. The cost is an integral functional, linear (but non-monotone) with respect to the state function. We prove the existence of optimal potentials and we provide some necessary conditions for optimality. Several numerical simulations are shown.