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

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 $q>1$.