Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $${\mathrm{\Delta }}^{2}u=u^{p}in{B}_1\mathrm{\backslash }\{0\}$$ with an isolated singularity, where the punctured ball $B_1 \backslash {0} \subset $\mathbb{R}$^n$with$n\geq 5$and$\frac{n}{n-4} < p < \frac{n+4}{n-4}$. This equation is relevant for the $Q$-curvature problem in conformal geometry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for $-\mathrm{\Delta }u$. We also give a new method to prove removable singularity theorem for nonlinear higher order equations.