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

Abstract

Of interest in this note is the following geometric equation, $\Delta^2 u + u^{-q} = 0$ in $\mathbf R^3$. 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 positive radially symmetric solution grows at least linearly and at most quadratically at infinity for any $q > 1$. In addition, when $q > 3$ any positive 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 positive 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 positive radially symmetric solutions of exactly given quadratic growth at infinity for any $q > 1$, hence completing the picture of positive radially symmetric solutions of the equation.