Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics

Abstract

We obtain the Bocher-type theorems and present the sharp characterization of the asymptotic behavior at the isolated singularities of solutions of some fourth and higher order equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in \begin{document}$ B \backslash \{0\} $\end{document} , where \begin{document}$ B \subset \mathbb{R}^N $\end{document} is the unit ball. The weights can be singular at \begin{document}$ x = 0 $\end{document} . We present the sharp asymptotic behavior of nonnegative solutions of the weighted elliptic equations near \begin{document}$ x = 0 $\end{document} and the Liouville-type results for the degenerate elliptic equations in \begin{document}$ \mathbb{R}^N \backslash \{0\} $\end{document} .