Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials

Let u be the weak solution to the degenerate Schrödinger equation with singular coefficients in Lipschitz domain as following −div w(x)A(x)∇u(x) +V (x)u(x)w(x) = 0, where A(x) is a real symmetric matrix function satisfying the elliptic condition and the Lipschitz continuity, w(x) is an A2 weight function ofMuckenhoupt class, and V (x) is the Fefferman–Phong’s potential. The weighted doubling properties and unique continuations for the weak solution u in the interior of any domains as well as at the boundary of some Lipschitz domains are derived in this paper. © 2007 Published by Elsevier Inc