Classification to the positive radial solutions with weighted biharmonic equation

In this paper, we consider the weighted problem Δ(|x|^-αΔu) = |x|^βu^p, u(x) > 0, u(x) = u(|x|) in ℝⁿ\{0}, where n ≥ 5,-n < α < n -4 and (p, α,β,n),p > 1 belongs to the critical hyperbola (Equation Presented). We give two type-homoclinic functions v(t) := |x| n-4-α/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 < α < n-4; while for -n < α ≤ -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 α = β = 0, it is related to the Q-curvature problem in conformal geometry.