This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{ equation*} \mathrm{ div}(|x|^{ \alpha}\nabla(\mathrm{ div}(|x|^\alpha\nabla u)))=u^{ 2^{ **}_{ \alpha}-1},\quad u>0\quad \mbox{ in}\quad \mathbb{ R}^N, \end{ equation*} where $N\geq 2$, $\frac{ 4-N}{ 2}<\alpha<2$ and $2^{ **}_{ \alpha}=\frac{ 2N}{ N-4+2\alpha}$. 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 $\alpha\neq 0$. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if $\alpha$ satisfies $(2-\alpha)(2N-2+\alpha)\neq4k(N-2+k)$ 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 $N\geq 5$ and $0<\alpha<2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality \begin{ equation*} \int_{ \mathbb{ R}^N} |\mathrm{ div}(|x|^\alpha\nabla u)|^2 \mathrm{ d}x \geq C \left(\int_{ \mathbb{ R}^N}|u|^{ 2^{ **}_{ \alpha}} \mathrm{ d}x\right)^{ \frac{ 2}{ 2^{ **}_{ \alpha}}},\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.
Categories: math.AP