Abstract:

In this paper, we consider the following second order variable coefficient weighted degenerate elliptic equation with singular potential constituted by Generalized Baouendi-Grushion vector fields

$$ - \sum\limits_{i,j = 1}^N {{X_j}({a_{ij}}(x,y){X_i}u)} + V(x,y)u = 0,$$

where

$$\lambda {\left| \eta \right|^2}w \leqslant \left\langle {A\eta,\eta } \right\rangle \leqslant {\lambda ^{ - 1}}{\left| \eta \right|^2}w,\ \ A = {({a_{ij}})_{N \times N}},$$

$w$ is a weight function related to quasi distance. By using the weighted Hardy inequality, Rellich type identity and the estimates of Dirichlet energy, we first get the monotonicity of the frequency function of the weak solutions. Then the doubling and unique continuation properties of the weak solutions are obtained.

Key words: unique continuation properties, Baouendi-Grushion vector fields, singular potential, frequency function