Strong unique continuation property for fourth order Baouendi-Grushin type subelliptic operators with strongly singular potential

Abstract

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation \begin{equation*} Δ^2_{X}u=Vu, \end{equation*} where ${\Delta }_X={\Delta }_x+|x{|}^{2\alpha }{\Delta }_y$ ($0<\alpha \le 1$), with $x\in {\mathbb{R}}^m,y\in {\mathbb{R}}^n$, denotes the Baouendi-Grushin type subelliptic operators, and the potential $V$ satisfies the strongly singular growth assumption $|V|\le \frac{c_0}{{\rho }^4}$, where \begin{equation*} ρ=\left(|x|^{2(α+1)}+(α+1)^2|y|^2\right)^{\frac{1}{2(α+1)}} \end{equation*} is the gauge norm. The main argument is to introduce an Almgren's type frequency function for the solutions, and show its monotonicity to obtain a doubling estimate based on setting up some refined Hardy-Rellich type inequalities on the gauge balls with boundary terms.

21pages