Existence and nonexistence of radial solutions for semilinear equations with bounded nonlinearities on exterior domains

Abstract

In this article we study radial solutions of Δu + K(r)ƒ(u) = 0 on the exterior of the ball of radius R > 0 centered at the origin in ℝN where ƒ is odd with ƒ < 0 on (0, β), ƒ > 0 on (β, δ), ƒ ≡ 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) = ∫u0 ƒ(t) dt has a "hilltop" at u = δ. With mild assumptions on ƒ we prove that if K(r) ~ r-α with 2 < α < 2(N - 1) then there are n solutions of Δu + K(r)ƒ(u) = 0 on the exterior of the ball of radius R such that u → 0 as r → ∞ if R > 0 is sufficiently small. We also show there are no solutions if R > 0 is sufficiently large.