In this paper we prove the existence of an infinite number of radial solutions of △u+K(r)f(u) =0 on the exterior of the ball of radius R ＞ 0 centered at the origin in ]RN where f is odd with f ＜ 0 on (0,β),f ＞ 0 on (β,δ),f ≡ 0 for u ＞ δ,and where the function K(r) is assumed to be positive and K(r) → 0 as r → ∞.The primitive F(u) =∫u0f(s) ds has a "hilltop" at u =δ which allows one to use the shooting method to prove the existence of solutions.