arXiv 2308.060142023-08math.AP

一类加权四阶方程正径向解的分类与非退化性及其应用

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

Shengbing Deng, Xingliang Tian

备注：: This version is the final one, corresponding to the paper now published in Nonlinear Analysis DOI: https://doi.org/10.1016/j.na.2023.113468

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AI总结：

本文研究一类加权四阶方程的正径向解，刻画其线性化问题解的结构与非退化条件，并应用于相关不等式余项分析、新型二阶Caffarelli-Kohn-Nirenberg不等式建立及指定扰动问题研究。

摘要：

本文研究如下加权四阶方程的径向解：\begin{equation*}\mathrm{div}(|x|^α\nabla(\mathrm{div}(|x|^α\nabla u)))=u^{2^{**}_α-1},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N,\end{equation*}其中$N\geq 2$，$\frac{4-N}{2}<α<2$，且$2^{**}_α=\frac{2N}{N-4+2α}$。显然，上述方程的解在缩放变换$λ^{\frac{N-4+2α}{2}}u(λx)$下具有不变性，但当$α\neq 0$时不具有平移不变性。我们刻画了径向解对应的线性化问题的所有解，得出结论：若$α$满足对所有$k\in\mathbb{N}^+$都有$(2-α)(2N-2+α)\neq4k(N-2+k)$，则径向解是非退化的；否则线性化问题存在“替代”平移不变性所对应解的新解。作为应用，首先我们研究了与上述方程相关的若干不等式的余项。随后当$N\geq 5$且$0<α<2$时，我们建立了一类新型的二阶Caffarelli-Kohn-Nirenberg不等式：\begin{equation*}\int_{\mathbb{R}^N} |\mathrm{div}(|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*}并在此情形下利用Lyapunov-Schmidt约化研究了一类指定扰动问题。

英文摘要：

This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{equation*} \mathrm{div}(|x|^α\nabla(\mathrm{div}(|x|^α\nabla u)))=u^{2^{**}_α-1},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 2$, $\frac{4-N}{2}<α<2$ and $2^{**}_α=\frac{2N}{N-4+2α}$. It is obvious that the solutions of above equation are invariant under the scaling $λ^{\frac{N-4+2α}{2}}u(λx)$ while they are not invariant under translation when $α\neq 0$. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if $α$ satisfies $(2-α)(2N-2+α)\neq4k(N-2+k)$ 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 $0<α<2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality \begin{equation*} \int_{\mathbb{R}^N} |\mathrm{div}(|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.

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