This article proves the existence of an infinite number of radial solutions of Δu+K(r)ƒ(u) = 0, one with exactly n zeros for each nonnegative integer n on the exterior of the ball of radius R > 0, Bʀ, centered at the origin in ℝᴺ with u = 0 on ∂Bʀ and limᵣ→∞u(r) = 0 where N > 2, f is odd with ƒ < 0 on (0; β), ƒ > 0 on (β;∞), ƒ(u) ~ uᵖ with 0 < p < 1 for large u and K(r) ~ r-α with 0 < α < 2 for large r.

Article proves the existence of an infinite number of radial solutions to ∆u+K(r)f(u) = 0 on ℝᶰ such that limᵣ →∞ u(r) = 0 with prescribed number of zeros on the exterior of the ball of radius R > 0 where f is odd with f < 0 on (0, β), f > 0 on (β, ∞) with f superlinear for large u, and K(r) ∼ r ⁻ᵅ with α > 2(N − 1).

This article proves the existence of an infinite number of radial solutions of Δ(u) + f(u) = 0 with prescribed number of zeros 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,∞) where β > 0.