Classification of positive radial solutions to a weighted biharmonic equation

<p style='text-indent:20px;'>In this paper, we consider the weighted fourth order equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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\}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 5 $\end{document}</tex-math></inline-formula>, id="M2">\begin{document}$ -n<\alpha<n-4 id="M3">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula> and id="M4">\begin{document}$ (p,\alpha,\beta,n) belongs to critical hyperbola</p><p id="FE2"> \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} n-2. style='text-indent:20px;'>We prove existence of radial solutions equation for some id="M5">\begin{document}$ \lambda id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. On other hand, let id="M7">\begin{document}$ v(t): |x|^{\frac{n-4-\alpha}{2}}u(|x|) id="M8">\begin{document}$ t -\ln |x| then solution id="M9">\begin{document}$ u with non-removable singularity at origin, id="M10">\begin{document}$ v(t) is a periodic function if id="M11">\begin{document}$ \alpha \in (-2,n-4) id="M12">\begin{document}$ id="M13">\begin{document}$ satisfy conditions; while id="M14">\begin{document}$ (-n,-2] there exists corresponding id="M15">\begin{document}$ not periodic. We also get results about best constant symmetry breaking, which closely related Caffarelli-Kohn-Nirenberg type inequality.</p>