Classification to the positive radial solutions with weighted biharmonic equation

In this paper, we consider the weighted problem

$ \Delta (|x|^{-\alpha} \Delta u) = |x|^{\beta} u^p, \qquad u(x)>0, \qquad u(x) = u(|x|)\qquad \text{in}\; \; \mathbb{R}^n\backslash{\{0}\}, $

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

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

We give two type-homoclinic functions $ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|), t = -\ln |x| $. On the other hand, for radial solution $ u $ with non-removable singularity at origin, $ v(t) $ is periodic and classification for all periodic functions are obtained with $ -2<\alpha<n-4 $; while for $ -n<\alpha \le -2, $ there always exists a solution $ u(|x|) $ with non-removable singularity and the corresponding function $ v(t) $ is not periodic. It is also closely related to the Caffarelli-Kohn-Nirenberg inequality, and we get some results such as the best embedding constants and the existence in radial case. In particular, for $ \alpha = \beta = 0 $, it is related to the $ Q $-curvature problem in conformal geometry.