  Our main task in this note is to prove the existence and to classify the
exact growth at infinity of radial positive $C^6$-solutions of $(-\Delta )^3 u
= u^p$ in $\mathbf{R}^n$, where $n\geqslant 15$ and $p$ is bounded from below
by the sixth-order Joseph-Lundgren exponent. Following the main work of
Winkler, we introduce the sub- and super-solution method and comparision
principle to conclude the asymptotic behavior of solutions.
