Multiple positive solutions to elliptic boundary blow-up problems

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,|x|<1,u(x)→∞,|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 toΔu+(a+(|x|)−μa−(|x|))g(u)=0,x∈RN, is also considered.