The fractional Schrödinger equation with singular potential and measure data

Abstract

We consider the steady fractional Schrödinger equation Lu+V u = f posed on a bounded domain Ω; L is an integro-differential operator, like the usual versions of the fractional Laplacian (−∆)s; V ≥ 0 is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of (−∆)s and prove well-posedness for functions as data. If V is bounded or mildly singular a unique solution of (−∆)su + V u = µ exists for every Borel measure µ. On the other hand, when V is allowed to be more singular, but only on a finite set of points, a solution of (−∆)su + V u = δx, where δx is the Dirac measure at x, exists if and only if h(y) = V (y)|x − y|−(n+2s) is integrable on some small ball around x. We prove that the set Z = {x ∈ Ω: no solution of (−∆)su + V u = δx exists} is relevant in the following sense: a solution of (−∆)su + V u = µ exists if and only if |µ|(Z) = 0. Furthermore, Z is the set points where the strong maximum principle fails, in the sense that for any bounded f the solution of (−∆)su + V u = f vanishes on Z.