The fractional Schr\"odinger equation with singular potential and measure data

Abstract

We consider the steady fractional Schr\"odinger equation $L u + V u = f$ posed on a bounded domain $\Omega$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ 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 $(-\Delta)^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-\Delta)^s u + V u = \mu$ exists for every Borel measure $\mu$. On the other hand, when $V$ is allowed to be more singular, but only on a finite set of points, a solution of $(-\Delta)^s u + V u = \delta_x$, where $\delta_x$ 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 $Z = \{x \in \Omega : \textrm{no solution of } (-\Delta)^s u + Vu = \delta_x \textrm{ exists}\}$ is relevant in the following sense: a solution of $(-\Delta)^s u + V u = \mu$ exists if and only if $|\mu| (Z) = 0$. Furthermore, $Z$ is the set points where the strong maximum principle fails, in the sense that for any bounded $f$ the solution of $(-\Delta)^s u + Vu = f$ vanishes on $Z$.