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arXiv · 1812.06555

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

Hui Yang
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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 $n\geq 5$ 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.

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Hui Yang. 2018-12-16. Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities. https://arxiv.org/abs/1812.06555

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