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

In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} Δ^2u=Vu, \end{equation*} where $V\in W^{1,\mathrm{\infty }}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(|V|^{\frac{1}{4}}{L^{\infty}}+|\nabla V|{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted 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 establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function.