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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 anyradially symmetric solution grows at least linearly and at most quadraticallyat infinity for any $q>1$. In addition, when $q>3$ any radially symmetricsolution is either exactly linear growth or exactly quadratic growth atinfinity. Recently, Guerra (J. Differential Equations 253, 3147-3157) has shownthat the equation always admits a unique radially symmetric solution of exactlygiven linear growth at infinity for any $q>3$ which is also necessary. In thisnote, by using the phase-space analysis, we show the existence of infinitelymany radially symmetric solutions of exactly given quadratic growth at infinityfor any $q>1$.

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