Existence and regularity of positive solutions of elliptic equations of Schrödinger type
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
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.
IdentifiersURN: urn:nbn:se:liu:diva-87816DOI: 10.1007/s11854-012-0045-zOAI: oai:DiVA.org:liu-87816DiVA: diva2:601842