In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $$ \Delta^2 u = u^p~~~~~~~in ~ B_1 \backslash \{ 0\}$$ 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 $-\Delta u$. We also give a new method to prove removable singularity theorem for nonlinear higher order equations.
Categories: math.AP