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 $${\Delta }^{2}u=u^{p}in{B}_1\backslash \{0\}$$ with an isolated singularity, where the punctured ball $B_1\backslash \{0\}\subset {\mathbb{R}}^n$ with $n\ge 5$ 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.

22 pages. This is a revision of arXiv:1812.06555v1. Some minor changes have been made