Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Isolated singularities of the Schrödinger equation with a good potential

Authors: Juan Luis Vázquez and Cecilia Yarur
Journal: Trans. Amer. Math. Soc. 315 (1989), 711-720
MSC: Primary 35J10; Secondary 35B05, 81C05
MathSciNet review: 932451
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Abstract: We study the behaviour near an isolated singularity, say 0, of nonnegative solutions of the Schrödinger equation $ - \Delta u + Vu = 0$ defined in a punctured ball $ 0 < \vert x\vert < R$. We prove that whenever the potential $ V$ belongs to the Kato class $ {K_n}$ the following alternative, well known in the case of harmonic functions, holds: either $ \vert x{\vert^{n - 2}}u(x)$ has a positive limit as $ \vert x\vert \to 0$ or $ u$ is continuous at 0. In the first case $ u$ solves the equation $ - \Delta u + Vu = a\delta $ in $ \{ \vert x\vert < R\} $. We discuss the optimality of the class $ {K_n}$ and extend the result to solutions $ u \ngeq 0$ of $ - \Delta u + Vu = f$.

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Keywords: Isolated singularities, Schrödinger equation, Kato potentials
Article copyright: © Copyright 1989 American Mathematical Society