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 $$\{\begin{array}{ll}\mathrm{\Delta }u+(a^{+}(\mathrm{\mid }x\mathrm{\mid })-\mu a^{-}(\mathrm{\mid }x\mathrm{\mid }))g(u)=0,& \mathrm{\mid }x\mathrm{\mid }<1,\\ u(x)\to \mathrm{\infty },& \mathrm{\mid }x\mathrm{\mid }\to 1,\end{array}$$ 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 $$\mathrm{\Delta }u+(a^{+}(\mathrm{\mid }x\mathrm{\mid })-\mu a^{-}(\mathrm{\mid }x\mathrm{\mid }))g(u)=0,x\in {\mathbb{R}}^N,$$ is also considered.