Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{equation*} $\mathrm{d}\mathrm{i}\mathrm{v}$(|x|^α\nabla($\mathrm{d}\mathrm{i}\mathrm{v}$(|x|^α\nabla u)))=u^{2^{}α-1},\quad u>0\quad \mbox{in}\quad $\mathbb{R}$^N, \end{equation*} where $N\ge 2$, $\frac{4-N}{2}<\alpha <2$ and $2_{\alpha }^{\ast \ast }=\frac{2N}{N-4+2\alpha }$. 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 $\alpha \mathrm{\ne }0$. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if $\alpha$ satisfies $(2-\alpha )(2N-2+\alpha )\mathrm{\ne }4k(N-2+k)$ 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 $N\ge 5$ and $0<\alpha <2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality \begin{equation*} \int{$\mathbb{R}$^N} |$\mathrm{d}\mathrm{i}\mathrm{v}$(|x|^α\nabla u)|^2 $\mathrm{d}$x \geq C \left(\int_{$\mathbb{R}$^N}|u|^{2^{}_α} $\mathrm{d}$x\right)^{\frac{2}{2^{**}_α}},\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.