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