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

In this paper, we consider polyharmonic Lane-Emden equations (-Delta)(m)u = vertical bar u vertical bar(p-1)u, in R-n, where m >= 3. We classify the stable or stable outside a compact set solutions when m = 3 or 4 for any dimensions and when m >= 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.