On the Strong unique continuation property of a degenerate elliptic operator with Hardy type potential

Abstract

In this paper we prove strong unique continuation for the following degenerate elliptic equation \begin{equation}\label{e0} \Delta_zu +|z|^2\partial_t^2u = Vu,\quad (z,t) \in \mathbb{R}^N \times \mathbb{R} \end{equation} where the potential $V$ satisfies either of the following growth assumptions \begin{align} & |V(z,t)| \leq \frac{f(\rho(z,t))}{\rho(z,t)^2},\ \text{where $f$ satisfies the Dini integrability condition as in (1.3)} \\ & \text{or when } \notag \\ & |V(z,t)| \leq C\frac{\psi(z,t)^{\epsilon}}{\rho(z,t)^2},\ \text{for some $\epsilon>0$ with $\psi$ as in (2.6) and $N$ even.} \notag \end{align} This extends some of the previous results obtained in [G] for this subfamily of Baouendi-Grushin operators. As corollaries, we obtain new unique continuation properties for solutions $u$ to \[ \Delta_{\mathbb{H}} u = Vu \] with certain symmetries as expressed in (1.6) where $\Delta_{\mathbb{H}}$ corresponds to the sub-Laplacian on the Heisenberg group $\mathbb{H}^n$.