The fractional Schrödinger equation with singular potential and measure data

Abstract

We consider the steady fractional Schrödinger equation $Lu+Vu=f$ 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 $(-\Delta {)}^{s}u+Vu=\mu$ 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 $(-\Delta {)}^{s}u+Vu={\delta }_x$, 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 $Z=\{x\in \Omega :\text{no solution of }(-\Delta {)}^{s}u+Vu={\delta }_x\text{ exists}\}$ is relevant in the following sense: a solution of $(-\Delta {)}^{s}u+Vu=\mu$ 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 $Z$.

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• The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions