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Existence and regularity of positive solutions of elliptic equations of Schrödinger type

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Abstract

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schrödinger type

$$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$

for an arbitrary open Ω ⊆ ℝn under only a form-boundedness assumption on σ ∈ D′(Ω) and ellipticity assumption on A ∈ L ∞(Ω)n×n. We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient

$$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$

As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schrödinger type operator H = −div(A∇·)-σ with arbitrary distributional potential σ ∈ D′(Ω), and give examples clarifying the relationship between these two properties.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    B. J. Jaye & I. E. Verbitsky

  2. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, UK

    V. G. Maz’ya

  3. Department of Mathematics, Linköping University, SE-581 83, Linköping, Sweden

    V. G. Maz’ya

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  1. B. J. Jaye
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  2. V. G. Maz’ya
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  3. I. E. Verbitsky
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Correspondence to B. J. Jaye.

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The first and third authors are supported in part by NSF grant DMS-0901550.

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Jaye, B.J., Maz’ya, V.G. & Verbitsky, I.E. Existence and regularity of positive solutions of elliptic equations of Schrödinger type. JAMA 118, 577–621 (2012). https://doi.org/10.1007/s11854-012-0045-z

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  • DOI: https://doi.org/10.1007/s11854-012-0045-z

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