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The fractional Schrödinger equation with singular potential and measure data

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  • We consider the steady fractional Schrödinger 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 : \rm{no solution of } (-\Delta)^s u + Vu = \delta_x \rm{ 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 $.

    Mathematics Subject Classification: 35R11, 35J10, 35D30, 35J67, 35J75.

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