Classification of positive radial solutions to a weighted biharmonic equation

In this paper, we consider the weighted fourth order equation

$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $

where $ n\geq 5 $, $ -n<\alpha<n-4 $, $ p>1 $ and $ (p,\alpha,\beta,n) $ belongs to the critical hyperbola

$ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $

We prove the existence of radial solutions to the equation for some $ \lambda $ and $ \mu $. On the other hand, let $ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|) $, $ t = -\ln |x| $, then for the radial solution $ u $ with non-removable singularity at origin, $ v(t) $ is a periodic function if $ \alpha \in (-2,n-4) $ and $ \lambda $, $ \mu $ satisfy some conditions; while for $ \alpha \in (-n,-2] $, there exists a radial solution with non-removable singularity and the corresponding function $ v(t) $ is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality.