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

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.