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

Abstract

We consider the biharmonic equation ${\Delta }^{2}u=u^{\alpha }$ in ${\mathbf{R}}^n$ with $n⩾1$. It was proved that this equation has a positive classical solution if, and only if, either $\alpha ⩽1$ with $n⩾1$ or $\alpha ⩾(n+4)/(n-4)$ with $n⩾5$. The asymptotic behavior at infinity of all positive radial solutions was known in the case $\alpha ⩾(n+4)/(n-4)$ and $n⩾5$. In this paper, we classify the asymptotic behavior at infinity of all positive radial solutions in the remaining case $\alpha ⩽1$ with $n⩾1$; hence obtaining a complete picture of the asymptotic behavior at infinity of positive radial solutions. Since the underlying equation is 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.

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