The authors consider the problem of finding a positive solution \(u\) of the differential equation \(\Delta u + K(|x|)u^p = 0\) in \(\mathbb{R}^n\setminus \{0\}\). Here \(K\) is a given function which is Hölder continuous in \(\mathbb{R}^n\setminus \{0\}\). Many authors (often in collaboration with Li) have studied this problem under various hypotheses on \(K\), and this paper is concerned with the case that \(K(r)\) behaves like \(r^l\) for some constant \(l >-2\). After pointing out that all solutions of this problem are radial, the authors examine solutions of the initial value problem \[ u'' + \frac {n-1}r u' +K(r)u^p =0,\;u(0) = \alpha, \] where \(\alpha\) is a positive parameter. Depending on the value of \(p\), either the solutions are monotone, that is, \(\alpha > \beta\) implies \(u_\alpha > u_\beta\), or \(u_\alpha - u_\beta\) switches signs infinitely many times. They also show that, under fairly general conditions, there is exactly one solution of the differential equation which is infinite at \(0\).