Existence of solutions for semilinear problems with prescribed number of zeros on exterior domains

In this article we prove the existence of an infinite number of radial solutions of $\Delta(u)+f(u)=0$ with prescribed number of zeros on the exterior of the ball of radius R>0 centered at the origin in $\mathbb {R}^N$ where f is odd with f<0 on $(0,\beta)$, f>0 on $(\beta,\infty)$ where $\beta>0$