Monotonicity Formula and Classification of Stable Solutions to Polyharmonic Lane–Emden Equations

Author:

Luo Senping1,Wei Juncheng2,Zou Wenming3

Affiliation:

1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

2. Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

3. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Abstract

Abstract In this paper, we consider polyharmonic Lane–Emden equations $$ \begin{equation*} (-\Delta )^m u=|u|^{p-1}u, \ \ \ \mbox{in} \ \ \ {\mathbb R}^n, \end{equation*}$$where $m\geq 3$. We classify the stable or stable outside a compact set solutions when $m=3$ or $4$ for any dimensions and when $m\geq 5$ for large dimensions. In the process, we exhibit the general Joseph–Lundgren exponent (including both local and nonlocal cases) in a concise form and prove related properties. The key ingredient of the proof of the classification is a monotonicity formula for general polyharmonic equations, which may have application in regularity theory for higher-order elliptic equations.

Funder

Natural Sciences and Engineering Research Council of Canada

National Natural Science Foundation of China

Jiangxi Double Thousand Plan

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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1. Classification of finite Morse index solutions to the polyharmonic Hénon equation;Calculus of Variations and Partial Differential Equations;2022-11-05

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