Weighted fourth order elliptic problems in the unit ball

Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball \begin{document}$ B $\end{document} are studied. The weights can be singular at \begin{document}$ x = 0 \in B $\end{document} . Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point \begin{document}$ x = 0 $\end{document} .