Asymptotic expansion of radial solutions for supercritical biharmonic equations

TL;DR

This paper derives the asymptotic expansion at infinity for positive radial solutions of the supercritical biharmonic equation, identifying stability conditions related to the critical power p_c.

Contribution

It provides the first detailed asymptotic expansion of solutions for p ≥ p_c, linking stability to the asymptotic behavior of solutions.

Findings

Asymptotic expansion at infinity for solutions when p ≥ p_c

Identification of the critical power p_c for stability

Connection between asymptotic behavior and linear stability

Abstract

Consider the positive, radial solutions of the nonlinear biharmonic equation $\Delta^2 u = u^p$. There is a critical power $p_c$ such that solutions are linearly stable if and only if $p\geq p_c$. We obtain their asymptotic expansion at infinity in the case that $p\geq p_c$.