Qualitative properties of weighted fourth order elliptic problems

Abstract

Qualitative properties of nonnegative solutions of weighted fourth order elliptic problems arising from the equations on singular manifolds with conical metrics are studied. The weights may be singular in the domains. We obtain the uniqueness of positive radial solution of the Navier boundary value problem in the ball $ B $ and its exact regularity at the singular point $ x = 0 $, which determines the exact regularity of the unique positive radial solution in $ B $. We also establish the Liouville type results for the equation on $ \mathbb{R}^N \backslash \{0\} $.