Existence of solutions for sublinear equations on exterior domains

In this article we consider the radial solutions of $\Delta u + K(r)f(u)= 0$ 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 $\frac{(N+2)-p(N-2)}{2} \leq \alpha < N-p(N-2)$ for large $r$. We prove existence of $n$ solutions - one with exactly $n$ zeros on $[R, \infty)$ - if $R>0$ is sufficiently small. If $R>0$ is sufficiently large then there are no solutions with $\lim_{r \to \infty} u(r)=0$