  We consider the biharmonic equation $\Delta^2 u = u^\alpha$ in $\mathbf R^n$
with $n \geqslant 1$. It was proved that this equation has a positive classical
solution if, and only if, either $\alpha \leqslant 1$ with $n \geqslant 1$ or
$\alpha\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at
infinity of all positive radial solutions was known in the case
$\alpha\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify
the asymptotic behavior at infinity of all positive radial solutions in the
remaining case $\alpha\leqslant 1$ with $n \geqslant 1$; hence obtaining a
complete picture of the asymptotic behavior at infinity of positive radial
solutions. Since the underlying equation is of higher order, we propose a new
approach which relies on a representation formula and asymptotic analysis
arguments. We believe that the approach introduced here can be conveniently
applied to study other problems with higher order operators.
