Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent

Abstract

Existence and uniqueness of positive radial solution \begin{document}$ u_p $\end{document} of the Navier boundary value problem: \begin{document}$ \left \{ \begin{array}{ll} \Delta^2 u = u^p \;\;\; &\mbox{in $ \mathbb{R} ^N \backslash {\overline B}$}, u>0 \;\;\; &\mbox{in $ \mathbb{R} ^N \backslash {\overline B}$}, u = \Delta u = 0 \;\;\; &\mbox{on $\partial B$}, \end{array} \right. $\end{document} where \begin{document}$ B \subset \mathbb{R} ^N \; (N \geq 5) $\end{document} is the unit ball and \begin{document}$ p>\frac{N+4}{N-4} $\end{document} , are obtained. Meanwhile, the asymptotic behavior as \begin{document}$ p \to \infty $\end{document} of \begin{document}$ u_p $\end{document} is studied. We also find the conditions such that \begin{document}$ u_p $\end{document} is non-degenerate.