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

TL;DR

This paper fully characterizes the asymptotic behavior at infinity of positive radial solutions to a biharmonic equation in all parameter regimes, using a novel approach based on representation formulas and asymptotic analysis.

Contribution

It provides a complete classification of asymptotic behaviors for all positive radial solutions of the biharmonic equation across all parameter ranges, introducing a new method applicable to higher order problems.

Findings

Classified asymptotic behavior for all solutions with $\alpha \\leq 1$

Extended understanding of solutions in the case $ \\geq 1$ and $\\alpha \\leq 1$

Introduced a new approach using representation formulas for higher order equations.

Abstract

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 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.