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,−div(w(x)A(x)∇u(x))+V(x)u(x)w(x)=0, where A(x)A(x) is a real symmetric matrix function satisfying the elliptic condition and the Lipschitz continuity, w(x)w(x) is an A2A2 weight function of Muckenhoupt class, and V(x)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.