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

In this paper, we prove the strong unique continuation property for the following fourth order degenerate elliptic equation Delta(2)(X)u =V u, where Delta(X) = Delta(X) + |x|(2 alpha) Delta y (0 < alpha <= 1), with x is an element of R-m, y is an element of R-n, denotes the Baouendi-Grushin type subelliptic operators, and the potential V satisfies the strongly singular growth assumption |V | < c(0)/rho(4) , where rho = |x|(2(alpha+1)) + (alpha + 1)(2)|y|(2))(1/2(alpha +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 HardyRellich type inequalities on the gauge balls with boundary terms. (c) 2023 Elsevier Inc. All rights reserved.