Classification of finite Morse index solutions to the polyharmonic Hénon equation
Classification of finite Morse index solutions to the polyharmonic Hénon equation
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DOI：
10.1007/s00526-022-02361-x
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发表时间：
2022-11
影响因子：
2.1
通讯作者：
Weiwei Ao;Shanshan Lai;S. Luo
中科院分区：
文献类型：
--
作者：
Weiwei Ao;Shanshan Lai;S. Luo
我们给出了多调和 Hénon 椭圆方程的稳定解和有限莫尔斯指数解的刘维尔型结果 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^mu=|x|^a|u|^{p-1}u,\quad \mathrm {in}\ {\mathbb {R}}^n, \end{aligned}$$\end{document}其中,,andnis 是一个大维度。为了构造齐次稳定解的分类定理，我们展示了临界指数的简洁而明确的形式，在取时称为约瑟夫-伦格伦指数。在此基础上，我们通过建立单调性公式、应用一些能量估计并最终使用排污分析来获得所需的结果。
We present Liouville-type results for stable solutions and finite Morse index solutions of the polyharmonic Hénon elliptic equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^mu=|x|^a|u|^{p-1}u,\quad \mathrm {in}\ {\mathbb {R}}^n, \end{aligned}$$\end{document}where,,andnis a large dimension. To construct the classification theorem of homogeneous stable solutions, we exhibit a concise and explicit form for a critical exponentwhich is known as the Joseph-Lundgren exponent when taking. Based on this, we obtain the desired results by establishing the monotonicity formula, applying some energy estimates and finally using a blow-down analysis.