Existence of solutions for sublinear equations on exterior domains

In this article we prove the existence of an infinite number of radial solutions of $\Delta u + K(r)f(u)= 0$, one with exactly n zeros for each nonnegative integer n on the exterior of the ball of radius R>0, $B_{R}$, centered at the origin in ${\mathbb R}^{N}$ with u=0 on $\partial B_{R}$ and $\lim_{r \to \infty} u(r)=0$ where N>2, f is odd with f<0 on $(0, \beta) $, f>0 on $(\beta, \infty)$, $f(u)\sim u^p$ with 0<p<1 for large u and $K(r) \sim r^{-\alpha}$ with $0 < \alpha < 2$ for large r