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)mu+V(x)u=0(−Δ)mu+V(x)u=0, m∈Nm∈N, V(x)V(x) is bounded. The bound shows that nontrivial solutions can not decay faster than e−|x|4/3ln⁡|x|e−|x|4/3ln⁡|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.