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

TL;DR

This paper classifies positive radial solutions of a weighted fourth-order PDE, analyzes their non-degeneracy, and applies findings to inequalities and perturbation problems, revealing new solution structures based on parameter conditions.

Contribution

It provides a complete classification of radial solutions, characterizes their linearized stability, and introduces a new second-order Caffarelli-Kohn-Nirenberg inequality with applications.

Findings

Radial solutions are explicitly characterized.

Non-degeneracy depends on parameter conditions.

New inequalities and perturbation results are established.

Abstract

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.