  Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation
$\Delta u + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type
theorem: if $u$ is a non-negative classical solution of the Lane-Emden
equation, then $u \equiv 0$. The method of moving planes combined with the
Kelvin transform provides an elegant proof of this theorem. A classical
approach of J. Serrin and H. Zou, originally used for the Lane-Emden system,
yields another proof but only in lower dimensions. Motivated by this, we
further refine this approach to find an alternative proof of the Liouville type
theorem in all dimensions.
