Removable singularity of the polyharmonic equation

Let x0∈Ω⊂Rnx0∈Ω⊂Rn, n≥2n≥2, be a domain and let m≥2m≥2. We will prove that a solution uu of the polyharmonic equation Δmu=0Δmu=0 in Ω∖{x0}Ω∖{x0} has a removable singularity at x0x0 if and only if |Δku(x)|=o(|x−x0|2−n)∀k=0,1,2,…,m−1 as |x−x0|→0|x−x0|→0 for n≥3n≥3 and =o(log(|x−x0|−1))∀k=0,1,2,…,m−1 as |x−x0|→0|x−x0|→0 for n=2n=2. For m≥2m≥2 we will also prove that uu has a removable singularity at x0x0 if |u(x)|=o(|x−x0|2m−n)|u(x)|=o(|x−x0|2m−n) as |x−x0|→0|x−x0|→0 for n≥3n≥3 and |u(x)|=o(|x−x0|2m−2log(|x−x0|−1))|u(x)|=o(|x−x0|2m−2log(|x−x0|−1)) as |x−x0|→0|x−x0|→0 for n=2n=2.