The fractional Schrödinger equation with singular potential and measure data

Abstract

We consider the steady fractional Schrödinger equation $L u + V u = f$ posed on a bounded domain $Ω$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-Δ)^s$; $V\ge 0$ 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 $(-Δ)^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-Δ)^s u + V u = μ$ 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 $(-Δ)^s u + V u = δ_x$, where $δ_x$ 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 $Z = \{x \in Ω: \textrm{no solution of } (-Δ)^s u + Vu = δ_x \textrm{ exists}\}$ is relevant in the following sense: a solution of $(-Δ)^s u + V u = μ$ 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 $(-Δ)^s u + Vu = f$ vanishes on $Z$.