We consider the biharmonic equation \(\varDelta ^2 u = u^\alpha \) in \(\textbf{R}^n\) with \(n \ge 1\) . It was proved that this equation has a positive classical solution if, and only if, either \(\alpha \le 1\) with \(n \ge 1\) or \(\alpha \ge (n+4)/(n-4)\) with \(n \ge 5\) . The asymptotic behavior at infinity of all positive radial solutions was known in the case \(\alpha \ge (n+4)/(n-4)\) and \(n \ge 5\) . In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case \(\alpha \le 1\) with \(n \ge 1\) ; 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 that 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.
