Existence and Nonexistence for Semilinear Equations on Exterior Domains

Abstract

In this paper we prove the existence of an infinite number of 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 $F(u)  = \int_{0}^{u} f(s) \, {\rm d}s$ has a  ``hilltop'' at $u=\delta$ which allows one to use the shooting method to prove the existence of solutions.