  In this paper, we study radial solutions of $\Delta u + K(|x|)f(u) = 0$ in
the exterior of the ball of radius $R > 0$ in $\mathbb{R}^N$ with $ N > 2$
where $f$ grows superlinearly at infinity and is singular at 0 with $f \sim
\frac{1}{|u|^{q-1}u}$ where $0 < q < 1.$ We also assume $ K(r) \sim |r|^{-
\alpha}$ for large $r$ and establish the existence of two infinite families of
solutions when $ N + q(N-2) < \alpha < 2(N-1).$
