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Existence and regularity of positive solutions of elliptic equations of Schrödinger type
Department of Mathematics, University of Missouri, Columbia, MO, 65211, United States, Department of Mathematics, Kent State University, Kent, OH, 44240, United States.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Department of Mathematics, University of Missouri, Columbia, MO, 65211, United States.
2012 (English)In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 118, no 2, 577-621 p.Article in journal (Refereed) Published
Abstract [en]

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. © 2012 Hebrew University Magnes Press.

Place, publisher, year, edition, pages
Springer Verlag (Germany) , 2012. Vol. 118, no 2, 577-621 p.
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URN: urn:nbn:se:liu:diva-87816DOI: 10.1007/s11854-012-0045-zOAI: diva2:601842
Available from: 2013-01-30 Created: 2013-01-23 Last updated: 2013-01-30

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Maz´ya, Vladimir
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Mathematics and Applied MathematicsThe Institute of Technology
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