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 equationΔX2u=Vu, where ΔX=Δx+|x|2αΔy (0<α≤1), with x∈Rm,y∈Rn, denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption |V|≤c0ρ4, whereρ=(|x|2(α+1)+(α+1)2|y|2)12(α+1) 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.