In this paper, we consider the weighted fourth order equation $$Δ(|x|^{-α}Δu)+λ\text{div}(|x|^{-α-2}\nabla u)+μ|x|^{-α-4}u=|x|^βu^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq 5$, $-n1$ and $(p,α,β,n)$ belongs to the critical hyperbola $$\frac{n+α}{2}+\frac{n+β}{p+1}=n-2.$$ We prove the existence of radial solutions to the equation for some $λ$ and $μ$. On the other hand, let $v(t):=|x|^{\frac{n-4-α}{2}}u(|x|)$, $t=-\ln |x|$, then for the radial solution $u$ with non-removable singularity at origin, $v(t)$ is a periodic function if $α\in (-2,n-4)$ and $λ$, $μ$ satisfy some conditions; while for $α\in (-n,-2]$, 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.

15 pages