On finite Morse index solutions to the quadharmonic Lane-Emden equation

Abstract

In this paper, we compute the Joseph-Lundgren exponent for the quadharmonic Lane-Emden equation, derive a monotonicity formula and classify the finite Morse index solution to the following quadharmonic Lane-Emden equation: \noindent \begin{equation}\nonumber Δ^4 u=|u|^{p-1}u\;\;\;\;\hbox{in}\;\;\;\;\; \R^n. \end{equation} As a byproduct, we also get a monotonicity formula for the quadharmonic maps $ Δ^4 u=0$.

53 pages; comments are welcome. arXiv admin note: text overlap with arXiv:1607.04719