  By using a suitable transform related to Sobolev inequality, we investigate
the sharp constants and optimizers in radial space for the following weighted
Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*}
\int_{\mathbb{R}^N}|x|^{\alpha}|\Delta u|^2 dx \geq
S^{rad}(N,\alpha)\left(\int_{\mathbb{R}^N}|x|^{-\alpha}|u|^{p^*_{\alpha}}
dx\right)^{\frac{2}{p^*_{\alpha}}}, \quad u\in C^\infty_c(\mathbb{R}^N),
\end{equation*} where $N\geq 3$, $4-N<\alpha<2$,
$p^*_{\alpha}=\frac{2(N-\alpha)}{N-4+\alpha}$. Then we obtain the explicit form
of the unique (up to scaling) radial positive solution $U_{\lambda,\alpha}$ to
the weighted fourth-order Hardy (for $\alpha>0$) or H\'{e}non (for $\alpha<0$)
equation: \begin{equation*}
  \Delta(|x|^{\alpha}\Delta u)=|x|^{-\alpha} u^{p^*_{\alpha}-1},\quad u>0 \quad
\mbox{in}\quad \mathbb{R}^N.
  \end{equation*} %Furthermore, we characterize all the solutions to the
linearized problem related to above equation at $U_{1,\alpha}$. For $\alpha\neq
0$, it is known the solutions of above equation are invariant for dilations
$\lambda^{\frac{N-4+\alpha}{2}}u(\lambda x)$ but not for translations. However
we show that if $\alpha$ is an even integer, there exist new solutions to the
linearized problem, which related to above equation at $U_{1,\alpha}$, that
"replace" the ones due to the translations invariance. This interesting
phenomenon was first shown by Gladiali, Grossi and Neves [Adv. Math. 249, 2013,
1-36] for the second-order H\'{e}non problem. Finally, as applications, we
investigate the reminder term of above inequality and also the existence of
solutions to some related perturbed equations.
