Existence and nonexistence of solutions for semilinear equations on exterior domains

In this article we study radial solutions of $\Delta u + K(r)f(u)= 0$ 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 $(\beta, \delta)$, $f\equiv 0$ for $u> \delta$, and where the function K(r) is assumed to be positive and $K(r)\to 0$ as $r \to \infty$. The primitive $F(u) = \int_0^u f(t) \, dt$ has a "hilltop" at $u=\delta$. We prove that if $K(r) \sim r^{-\alpha}$ with $\alpha> 2(N-1)$ and if R>0 is sufficiently small then there are a finite number of solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R such that $u \to 0$ as $r \to \infty$. We also prove the nonexistence of solutions if R is sufficiently large