The authors study positive solutions of the equation \[ \Delta^2 u + u^{-q} = 0 \quad\text{in}\quad {\mathbb R}^3, \leqno(1) \] where \(q>0\) is a constant. This equation arises in conformal geometry in the following way. Given a smooth Riemannian manifold \((M,g)\) with \(n=\text{dim}\, M \geq 3\), the Paneitz operator is defined by \[ P_g = \Delta_g^2 - \delta[ (a_n R_g g + b_n \text{Ric}_g) d] + {{n-4}\over{2}} Q_g, \] where \(Q_g\) is a specific function depending on \(n\), \(\Delta R_g\), \(R_g\) and \(\text{Ric}_g\), and \(a_n\), \(b_n\) are certain constants depending only on \(n\). This operator has the property that if \(\bar g= u^{4/(n-4)}g\), \(n\neq 4\), \(u\in C^\infty(M)\) positive, is a metric conformal to \(g\), then for any \(\varphi\in C^\infty(M)\), \[ P_g(\varphi u) = u^{{n+4}\over{n-4}} P_{\bar g}\varphi. \] In particular, if \(\varphi\equiv 1\), then \[ P_g u = {{n-4}\over{2}} Q_g u^{{n+4}\over{n-4}}. \leqno(2) \] A key step in understanding \((2)\) is to understand the simplest case that \((M,g)\) is the standard metric on \({\mathbb R}^n\). In this case \((2)\) simplifies to \[ \Delta^2 u = {{n-4}\over{2}} Q_{\bar g} u^{{n+4}\over{n-4}} \quad\text{in}\quad {\mathbb R}^n. \leqno(3) \] For \(n\geq 5\) and \(Q_{\bar g}\) being a positive constant, smooth positive solutions of \((3)\) have been classified by \textit{C.-S.~Lin} [Comment. Math. Helv. 73, No. 2, 206--231 (1998; Zbl 0933.35057)], and \textit{X.~Xu} [Pac. J. Math. 225, No. 2, 361--378 (2006; Zbl 1136.35039); Proc. R. Soc. Edinb., Sect. A, Math. 130, No. 3, 651--670 (2000; Zbl 0961.35037)]. The case \(n=3\) is significantly different from the case \(n\geq 5\) because for \(n=3\) the exponent in \((3)\) is negative. The main results proved are that if \((1)\) admits a smooth positive solution on \({\mathbb R}^3\), then \(q>1\). Furthermore, if \(13\), then \((1)\) has smooth positive solutions with exactly quadratic growth at infinity (these correspond to incomplete conformal metrics on \({\mathbb S}^3\) with \(Q\equiv 1\)). Moreover, all radially symmetric smooth positive solutions are either of exactly quadratic growth or exactly linear growth at infinity.