EXISTENCE AND REGULARITY OF POSITIVE SOLUTIONS OF ELLIPTIC EQUATIONS OF SCHRODINGER TYPE

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schrodinger type -div(A del u) - sigma u = 0 in Omega for an arbitrary open Omega subset of R-n under only a form-boundedness assumption on sigma is an element of D' (Omega) and ellipticity assumption on A is an element of L-infinity(Omega)(nxn). 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 -div(A del nu) = (A del nu) center dot del nu + sigma in Omega. As a consequence, we obtain necessary and sufficient conditions for both form-boundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schrodinger type operator H = -div(A del center dot) - sigma with arbitrary distributional potential sigma is an element of D'(Omega), and give examples clarifying the relationship between these two properties.