Two Infinite Families of Solutions for Singular Superlinear Equations on Exterior Domains

In this paper, we study radial solutions of $\mathrm{\Delta }u+K(\mathrm{\mid }x\mathrm{\mid })f(u)=0$ 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 $f\sim \frac{1}{\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u}$ where $0<q<1.$ We also assume $K(r)\sim \mathrm{\mid }r{\mathrm{\mid }}^{-\alpha }$ for large $r$ and establish the existence of two infinite families of solutions when $N+q(N-2)<\alpha <2(N-1)\mathrm{.}$