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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 classicalsolution 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 atinfinity of all positive radial solutions was known in the case$\alpha\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classifythe asymptotic behavior at infinity of all positive radial solutions in theremaining case $\alpha\leqslant 1$ with $n \geqslant 1$; hence obtaining acomplete picture of the asymptotic behavior at infinity of positive radialsolutions. Since the underlying equation is of higher order, we propose a newapproach which relies on a representation formula and asymptotic analysisarguments. We believe that the approach introduced here can be convenientlyapplied to study other problems with higher order operators.

A complete description of the asymptotic behavior at infinity of positive radial solutions to $\Delta^2 u = u^\alpha$ in $\mathbf R^n$