Quantitative uniqueness of some higher order elliptic equations.

We study the quantitative unique continuation property of some higher order elliptic operators. We prove a lower bound for nontrivial solutions of the equation ( − Δ ) m u + V ( x ) u = 0 , m ∈ N , V ( x ) is bounded. The bound shows that nontrivial solutions can not decay faster than e − | x | 4 / 3 ln ⁡ | x | at infinity. Moreover, we obtain an improved lower bound of nontrivial solutions for a special forth order elliptic operators in dimension 2, the bound is shown to be essentially sharp by constructing a Meshkov-type example. [ABSTRACT FROM AUTHOR]