  This paper is devoted to radial solutions of the following weighted
fourth-order equation
  \begin{equation*}
  \mathrm{div}(|x|^{\alpha}\nabla(\mathrm{div}(|x|^\alpha\nabla
u)))=u^{2^{**}_{\alpha}-1},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N,
  \end{equation*} where $N\geq 2$, $\frac{4-N}{2}<\alpha<2$ and
$2^{**}_{\alpha}=\frac{2N}{N-4+2\alpha}$. It is obvious that the solutions of
above equation are invariant under the scaling
$\lambda^{\frac{N-4+2\alpha}{2}}u(\lambda x)$ while they are not invariant
under translation when $\alpha\neq 0$. We characterize all the solutions to the
related linearized problem about radial solutions, and obtain the conclusion of
that if $\alpha$ satisfies $(2-\alpha)(2N-2+\alpha)\neq4k(N-2+k)$ for all
$k\in\mathbb{N}^+$ the radial solution is non-degenerate, otherwise there exist
new solutions to the linearized problem that ``replace'' the ones due to the
translations invariance. As applications, firstly we investigate the remainder
terms of some inequalities related to above equation. Then when $N\geq 5$ and
$0<\alpha<2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg
inequality
  \begin{equation*}
  \int_{\mathbb{R}^N} |\mathrm{div}(|x|^\alpha\nabla u)|^2 \mathrm{d}x \geq C
\left(\int_{\mathbb{R}^N}|u|^{2^{**}_{\alpha}}
\mathrm{d}x\right)^{\frac{2}{2^{**}_{\alpha}}},\quad
  \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N),
  \end{equation*}
  and in this case we consider a prescribed perturbation problem by using
Lyapunov-Schmidt reduction.
