We prove the existence of multiple positive radial solutions to the
sign-indefinite elliptic boundary blow-up problem {Δu+(a+(∣x∣)−μa−(∣x∣))g(u)=0,u(x)→∞,∣x∣<1,∣x∣→1, 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 μ>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 Δu+(a+(∣x∣)−μa−(∣x∣))g(u)=0,x∈RN, is also considered