Schr\"odinger equations with smooth measure potential and general measure data

Abstract

We study equations driven by Schr\"odinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures absolutely continuous with respect to a capacity generated by the operator. In particular, we cover positive potentials exploding on a set of capacity zero. The right-hand side of equations is allowed to be a general bounded Borel measure. The class of self-adjoint Dirichlet operators is quite large. Examples include integro-differential operators with the local part of divergence form. We give a necessary and sufficient condition for the existence of a solution, and prove some regularity and stability results.