Quantitative unique continuation property for solutions to a bi‐Laplacian equation with a potential

Abstract In this paper, we focus on the quantitative unique continuation property of solutions to where . We show that the maximal vanishing order of the solutions is not larger than where constant depends only and . Our key argument is to lift the original equation to that with a positive potential, then decompose the resulting fourth‐order equation into a special system of two second‐order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establish some doubling inequalities with explicit dependence on the Sobolev norm of the potential function. It is worth mentioning that the exponent of is optimal based on the analogues of the eigenfunctions of the bi‐Laplacian.