Strong unique continuation for $m$-th powers of a Laplacian operator with singular coefficients

In this paper we prove strong unique continuation for $u$ satisfying an inequality of the form $\vert\triangle^m u\vert \leq f(x,u,Du,\cdots,D^ku)$, where $k$ is up to $[3m/2]$. This result gives an improvement of a work by Colombini and Grammatico (1999) in some sense. The proof of the main theorem is based on Carleman estimates with three-parameter weights $\vert x\vert^{2\sigma_1}(\log\vert x\vert)^{2\sigma_2}\exp (\frac{\beta}{2}(\log \vert x\vert)^2)$.