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

Abstract

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $$\begin{aligned} \Delta ^2 u = u^p\quad \text{ in }\; B_1\backslash \{0\} \end{aligned}$$with an isolated singularity, where the punctured ball \(B_1 \backslash \{0\} \subset {\mathbb {R}}^n\) with \(n\ge 5\) 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.