  We prove the existence of multiple positive radial solutions to the
sign-indefinite elliptic boundary blow-up problem \[ \left\{\begin{array}{ll}
\Delta u + \bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, &
\; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array}
\right. \] where $g$ is a function superlinear at zero and at infinity, $a^+$
and $a^-$ are the positive/negative part, respectively, of a sign-changing
function $a$ and $\mu > 0$ is a large parameter. In particular, we show how the
number of solutions is affected by the nodal behavior of the weight function
$a$. The proof is based on a careful shooting-type argument for the equivalent
singular ODE problem. As a further application of this technique, the existence
of multiple positive radial homoclinic solutions to $$ \Delta u +
\bigl(a^+(\vert x \vert) - \mu a^-(\vert x \vert)\bigr) g(u) = 0, \qquad x \in
\mathbb{R}^N, $$ is also considered.
