On the existence of optimal potentials on unbounded domains

We consider elliptic equations of Schrödinger type with a right-hand side fixed and with the linear part of order zero given by a potential $V$. The main goal is to study the optimization problem for an integral cost depending on the solution $u_V$, when $V$ varies in a suitable class of admissible potentials. These problems can be seen as the natural extension of shape optimization problems to the framework of potentials. The main result is an existence theorem for optimal potentials, and the main difficulty is to work in the whole Euclidean space $R^d$, which implies a lack of compactness in several crucial points. In the last section we present some numerical simulations.