A note on radial solutions of $\Delta^2 u + u^{-q} = 0$ in $\mathbb R^3$ with exactly quadratic growth at infinity

TL;DR

This paper investigates radially symmetric solutions to a biharmonic equation with a negative power nonlinearity in three dimensions, establishing the existence of infinitely many solutions with quadratic growth at infinity using phase-space analysis.

Contribution

It demonstrates the existence of infinitely many radially symmetric solutions with quadratic growth at infinity for all q>1, extending previous results on linear growth solutions.

Findings

Existence of infinitely many solutions with quadratic growth at infinity.

Solutions are characterized using phase-space analysis.

Results hold for all q>1.

Abstract

Of interest in this note is the following geometric interesting equation $\Delta^2 u + u^{-q} = 0$ in $\mathbb R^3$. It was found by Choi-Xu (J. Differential Equations 246, 216-234) and McKenna-Reichel (Electron. J. Differential Equations 37 (2003)) that the condition $q>1$ is necessary and any radially symmetric solution grows at least linearly and at most quadratically at infinity for any $q>1$. In addition, when $q>3$ any radially symmetric solution is either exactly linear growth or exactly quadratic growth at infinity. Recently, Guerra (J. Differential Equations 253, 3147-3157) has shown that the equation always admits a unique radially symmetric solution of exactly given linear growth at infinity for any $q>3$ which is also necessary. In this note, by using the phase-space analysis, we show the existence of infinitely many radially symmetric solutions of exactly given quadratic growth at infinity for any $q>1$.