The differential equation under consideration is (1) \(y''=fy^{- \lambda}\) in an interval \([t_ 0,\infty)\), where \(\lambda\) is a positive constant and \(f\) is a real-valued continuous function in \([t_ 0,\infty)\) such that \(\hbox{supp}f(t)\) is not compact. Various conditions on \(f(t)\) are given which guarantee the existence or uniqueness of a positive solution \(y(t)\) of (1) such that both \(y(t)\) and \(y'(t)\) have limit zero at \(\infty\). In particular, sufficient conditions for the existence of such a solution are the convergence and nonnegativity of \(A(t)=\int^ \infty_ t f(s)ds\) on \([t_ 0,\infty)\) and the finiteness of \[ \int^ \infty_{t_ 0}t| f(t)|\left[\int^ \infty_ t A(s)ds\right]^{-\lambda/(\lambda+1)}dt. \] The methods include a variant of the shooting method and the technique of \textit{M. Naito} [Trans. Am. Math. Soc. 282, 577-588 (1984; Zbl 0556.34055)].